Asymptotics of the Divisor for the Good Boussinesq Equation

Abstract

We consider a third-order operator under the three-point Dirichlet condition. Its spectrum is the so-called auxiliary spectrum for the good Boussinesq equation, as well as the Dirichlet spectrum for the Schrödinger operator on the unit interval is the auxiliary spectrum for the periodic KdV equation. The auxiliary spectrum is formed by projections of the points of the divisor onto the spectral plane. We estimate the spectrum and the corresponding norming constants in terms of small operator coefficients. This work is the first in a series of papers devoted to solving the inverse problem for the Boussinesq equation.

Keywords

third-order operators divisor multipoint problems auxiliary spectrum asymptotics

1. Introduction and Main Results

1.1. Introduction

The operator $\partial^{3} + p \partial + \partial p + q, \partial = \partial / \partial x$ , with periodic $p$ and $q$ , is the Lax operator for the periodic good Boussinesq equation (below the GBE)

p_{t t} = - \frac{1}{3} (p_{x x x x} + 4 (p^{2})_{x x}), p_{t} = q_{x},

see McKean (1981). In order to solve the GBE, we need spectral analysis of this operator. Solutions of the GBE can be described in terms of two objects. The first one is the Riemann surface of the Lax operator. It is invariant with respect to the Boussinesq flow. The second one is a set of points on a Riemann surface, a divisor. It parameterizes solutions to the Boussinesq equation. The projections of the points of the divisor onto the spectral plane form the auxiliary spectrum of the Lax operator.

We discuss our plan for studying the inverse spectral problem for the Boussinesq equation on a circle. The experience with the Hill operator Korotyaev (1999) shows us that we need three sets of spectral data.

The set of branch points of the Riemann surface for the operator, this set coincides with the spectrum of the 2-periodic problem,

The auxiliary spectrum, which coincides with the spectrum of the Dirichlet problem.

The sequence $(s_{n})_{n \in N}$ of signs $s_{n} = +$ or $s_{n} = -$ , i.e. the sign of the norming constants.

The branch points of the Riemann surface are the edges of the gaps in the spectrum of the operator on the line. Each finite gap contains exactly one eigenvalue of the Dirichlet problem. Moreover, when the potential is shifted by a period, the eigenvalue of the Dirichlet problem, which lies in the n-th gap, runs through it monotonically from one edge to the other, making exactly n revolutions. Note that in Marchenko and Ostrovskii (1975) (see also Korotyaev, 1997, where the more simple proof is given) the authors work with the Riemann surface in terms of the quasimomentum, which is more complicated.

The situation for the Boussinesq equation is similar to that for the KdV equation, see Badanin and Korotyaev (2024b). In order to uniquely define the coefficients $p, q$ we need 3 sets of the spectral data.

The set of branch points of the Riemann surface for the Lax operator. The branch points, with exception of a finite number, are real, while the 2-periodic spectrum asymptotically lies near the line parallel to the imaginary axis.

The auxiliary spectrum coincides with the spectrum of the 3-point Dirichlet problem.

The sequence $(s_{n})_{n \in Z ∖ {0}}$ of signs $s_{n} = +$ or $s_{n} = -$ , i.e. the sign of the norming constants.

In our case of a third-order operator, the set of branch points is not the spectrum of any problem associated with the operator. But it is still real, with the possible exception of a finite number of points. There are some intervals on the Riemann surface, ‘pseudo-gaps’, similar to the gaps in the spectrum of the Hill operator. Edges of these intervals are adjacent branch points. The 3-point Dirichlet eigenvalues, with the exception of a finite number, are real, each ‘pseudo-gap’ contains exactly one eigenvalue, and when the potential is shifted by a period, the eigenvalue of the problem lying in the n-th ‘pseudo-gap’ runs monotonically from one edge to the other, making exactly n turns.

Moreover, there is an important relation between the spectral data for the third-order operator and the spectral data for the Hill operator. In fact, McKean (1981) introduced a transformation of the third-order equation with 1-periodic coefficients to the Hill equation with an energy-dependent potential. He showed that under this transformation the set of ramifications turns into a set of eigenvalues of the 2-periodic problem for the Schrödinger operator. Additionally, in Badanin and Korotyaev (2024c) we show that under McKean’s transformation the set of the eigenvalues of the 3-point Dirichlet problem for the 3-order operator turns into the set of the eigenvalues of the Dirichlet problem for the Schrödinger operator.

In general, some eigenvalues of the 3-point problem may be non-real and have multiplicity $⩾ 2$ . However, if $p$ and $q$ are small, then all eigenvalues are real and simple. In Badanin and Korotyaev (2024a) we construct the mapping from the space of coefficients $p, q$ onto the space of spectral data and prove, that this mapping is a real analytic bijection between the neighborhood of the point $p = q = 0$ onto the image of this neighborhood. Ultimately, this leads to the fact that a solution of the Boussinesq equation with sufficiently small initial data from $L^{2} (T)$ exists globally in this class. In the case when the coefficients of the operator are not small, the eigenvalues can be non-real and multiples. This greatly complicates the analysis and leads to the fact that the solutions of the Boussinesq equation have a blow-up discovered by Kalantarov and Ladyzhenskaja (1977). The corresponding inverse problem still needs to be solved.

Our article is devoted to the auxiliary spectrum and the corresponding norming constants for the Boussinesq equation. We consider a non-self-adjoint operator $L = L (u)$ acting on $L^{2} (0, 2)$ and given by

L y = (y^{″} + p y)^{'} + p y^{'} + q y, y (0) = y (1) = y (2) = 0,

(1.1)

where the 1-periodic coefficients

p, q

satisfy

u = (p, q) \in H = H \oplus H .

(1.2)

Here

H = L_{R}^{2} (T), T = R / Z

, is the Hilbert space, equipped with the norm

‖ f ‖^{2} = \int_{0}^{1} | f (x) |^{2} d x

. The spectrum of

L

is pure discrete (Badanin & Korotyaev, 2021) and it is complex.

The main goal of the present work is to obtain energy-uniform estimates for perturbation of the spectrum and the norming constants of the 3-point problem in the case of small coefficients. These results are necessary for solving the inverse 3-point problem, which will be the topic of our next work.

McKean (1981) considered the Lax operator of the periodic GBE with the coefficients $p, q \in C^{\infty} (T)$ . For the case of small coefficients he obtained some uniqueness results in the inverse spectral problem. His paper contains many important ideas and results, but his arguments cannot be directly applied to the case of non-smooth coefficients. In our paper Badanin and Korotyaev (2021) we considered the operator $L$ , given by (1.1), with the coefficients $p, q \in L^{1} (T)$ . We determined its eigenvalue asymptotics at high energy and the trace formula for this operator. Unfortunately, it is not enough to solve the inverse spectral 3-point problem. We need sharp uniform asymptotics of eigenvalues and the norming constants for small coefficients, which are determined here. Note that we are not aware of any works that discuss norming constants for higher-order operators with multipoint conditions and where similar uniform estimates were obtained. Apparently, our results in these directions are completely new.

The Lax operator for the bad periodic Boussinesq equation is self-adjoint third-order operators with periodic coefficients. Let us describe briefly results about it. The spectrum is absolutely continuous and covers the real line. It has multiplicity 1 or 3. The spectrum of multiplicity 3 consists of a finite number $⩾ 0$ of bounded intervals separated by gaps. Edges of the gaps are branch points of the Riemann surface. The branch points are asymptotically located far from the real axis. A detailed study of the spectral properties of these operators was begun by the authors in Badanin and Korotyaev (2012), Badanin and Korotyaev (2014a). We analyzed the Riemann surface and the spectrum and determined high energy asymptotics of the branch points and the 2-periodic spectrum. Moreover, we considered the case of small coefficients. In the case of small coefficients either whole spectrum has multiplicity one or there is exactly one small interval with the spectrum of multiplicity 3. The asymptotics of this small interval was determined in Badanin and Korotyaev (2012). Amour (1999), Amour (2001) considered self-adjoint third-order operators on the interval $[0, 1]$ under the Dirichlet plus quasi-periodic boundary conditions. He proved that two spectra uniquely define the operator and obtained explicit formulas for isospectral flows.

Higher-order operators and operators with matrix periodic coefficients have numerous applications. For this reason, they have been the subject of research by mathematicians for many years. Gelfand (1950) proved the decomposition of periodic operators on the line into a direct integral. These operators, in different classes of coefficients, have been considered by Badanin and Korotyaev (2014b), Badanin and Korotyaev (2015), Mikhailets and Molyboga (2004), Papanicolaou (1995), Papanicolaou (2003), $\dots$ The general theory of linear systems of differential equations with periodic coefficients was developed by Gel’fand and Lidskii (1955), Krein (1983) and Chelkak and Korotyaev (2006b), Korotyaev (2010), see also Carlson (2000), Clark et al. (2000). Korotyaev with co-authors consider the matrix Hill operators (Badanin et al., 2006), Chelkak and Korotyaev (2006b), and the matrix periodic Dirac type operators (Korotyaev, 2008), Korotyaev (2010), Korotyaev (2020). There the Riemann surface and Lyapunov functions were analyzed, and based on this analysis the spectral properties of the operator were studied. Moreover, we constructed a conformal mapping (averaged quasimomentum) and obtained various properties of this mapping. We defined various new trace formulas for the potentials and the Lyapunov exponent and obtained a priori estimates of the gap lengths in terms of the potentials. Finally, we note that the inverse problem for the Schrödinger operator with matrix-valued potential was solved by Chelkak and Korotyaev in Chelkak and Korotyaev (2006a), Chelkak and Korotyaev (2009) via eigenvalues plus matrix-valued norming constants.

There is a huge number of articles on spectral asymptotics for higher order operators. Multipoint problems are much less studied, see the short review in our paper Badanin and Korotyaev (2021). At the same time, apparently, such problems are important in the theory of nonlinear completely integrable systems.

Note that the spectral analysis of scalar higher-order operators with periodic coefficients and the periodic systems (the first and the second order) has both common properties and fundamentally different ones, see Badanin et al. (2006), Chelkak and Korotyaev (2006b), Korotyaev (2008), Korotyaev (2010), Badanin and Korotyaev (2011). Fundamental solutions of the periodic system are uniformly bounded on the real line, while higher-order equations have exponentially increasing fundamental solutions. It is necessary to take into account the contribution of bounded solutions to the spectral asymptotics against the background of the contribution of increasing solutions. This requires a more complicated analysis than the corresponding analysis for the Hill operators. At the same time, the Riemann surfaces of the matrix Hill operators at high energy are more complex than the Riemann surfaces of higher-order operators with periodic coefficients. This complicates the spectral analysis of the matrix operators.

The difficulties we encounter in analyzing the operator $L$ are due to two circumstances. First, we are dealing with a higher-order operator. As has already been said, in order to determine spectral asymptotics for such operators we have to use more complicated techniques compared to second-order operators. Second, $L$ is a non-self-adjoint operator. For this reason, we must simultaneously analyze two differential equations: the direct one and the transposed one, which corresponds to the formally adjoint differential expression.

1.2. Eigenvalues

Introduce the fundamental solutions $φ_{1}, φ_{2}, φ_{3}$ of the equation:

(y^{″} + p y)^{'} + p y^{'} + q y = λ y, λ \in C,

(1.3)

satisfying the conditions

φ_{j}^{[k - 1]} |_{x = 0} = δ_{j k}, j, k = 1, 2, 3

, where

y^{[0]} = y, y^{[1]} = y^{'}, y^{[2]} = y^{″} + p y

. The spectrum

σ (L)

L

is pure discrete and satisfies

σ (L) = {λ \in C : Δ (λ) = 0},

(1.4)

where the characteristic function

Δ

is the entire function given by

Δ (λ) = det (\begin{matrix} φ_{2} (1, λ) & φ_{3} (1, λ) \\ φ_{2} (2, λ) & φ_{3} (2, λ) \end{matrix}) .

(1.5)

All eigenvalues of

L

p = q = 0

are simple, real, and have the form:

μ_{n}^{o} = (z_{n}^{o})^{3}, where z_{n}^{o} = \frac{2 π n}{\sqrt{3}}, n \in Z_{0} = Z ∖ {0} .

(1.6)

Introduce the Hilbert spaces:

H^{1} = H^{1} (T) = {f, f^{'} \in L_{R}^{2} (T)}, H^{1} = H^{1} \oplus H,

(1.7)

equipped with the norm:

‖ u ‖_{1}^{2} = ‖ p ‖^{2} + ‖ p^{'} ‖^{2} + ‖ q ‖^{2},

(1.8)

respectively. Introduce the ball

B_{1} (r) \subset H^{1}

B_{1} (r) = {u \in H^{1} : ‖ u ‖_{1} 0.

(1.9)

Introduce the Fourier coefficients:

{\hat{f}}_{n} = \int_{0}^{1} f (x) e^{- i 2 π n x} d x, {\hat{f}}_{c n} = \int_{0}^{1} f (x) \cos 2 π n x d x, {\hat{f}}_{s n} = \int_{0}^{1} f (x) \sin 2 π n x d x, n \in Z .

Introduce the domains

D_{n}

D_{n} = {λ \in C : | z - z_{n}^{o} | < 1}, D_{- n} = - D_{n}, n ⩾ 0,

(1.10)

here and below

z = λ^{\frac{1}{3}}, \arg z \in (- \frac{π}{3}, \frac{π}{3}], \arg λ \in (- π, π] .

Throughout the text below,

ε > 0

is some fixed, sufficiently small number. In addition, we will denote by

C

various positive constants whose values depend only on

ε

. We formulate our first main result.

Theorem 1.1

Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then there is exactly one simple real eigenvalue $μ_{n}$ of the operator $L$ in each domain $D_{n}, n \in Z_{0} = Z ∖ {0}$ , which satisfies:

| μ_{n} - μ_{n}^{o} + a_{n} - \frac{b_{n}}{\sqrt{3}} | ⩽ \frac{C ‖ u ‖_{1}^{2}}{n},

(1.11)

for some

C > 0

, where

a_{n} = \frac{{\hat{p}}_{s n}^{'}}{\sqrt{3}} + {\hat{q}}_{c n}, b_{n} = \frac{{\hat{p}}_{c n}^{'}}{\sqrt{3}} - {\hat{q}}_{s n} .

(1.12)

Remark

In Badanin and Korotyaev (2021) we proved that if $u \in H^{1}$ , then

μ_{n} = μ_{n}^{o} - \frac{1}{\sqrt{3}} ({\hat{p}}_{s n}^{'} - \frac{{\hat{p}}_{c n}^{'}}{\sqrt{3}}) - ({\hat{q}}_{c n} + \frac{{\hat{q}}_{s n}}{\sqrt{3}}) + O (n^{- 1 / 2}), n \to \pm \infty .

The relation (1.11) improves this result and gives a uniform estimate for small

p, q

1.3. Norming Constants

The eigenvalues $μ_{n}, n \in Z_{0}$ , do not uniquely determine the coefficients $p$ and $q$ . To recover the coefficients, it is necessary to add the set of norming constants.

Recall the norming constants for the Schrödinger operator from Pöschel and Trubowitz (1987). Consider the operator $- y^{″} + V y$ on $L^{2} (0, 1)$ under the Dirichlet boundary conditions $y (0) = y (1) = 0$ , where $V \in H$ , $\int_{0}^{1} V (x) d x = 0$ . Recall that the spectrum consists of the simple eigenvalues $m_{1} < m_{2} < \dots$ , $m_{n} = (π n)^{2} + O (1)$ , as $n \to + \infty$ . These eigenvalues are zeros of an entire function $φ (1, \cdot)$ , where $φ (x, E)$ is the fundamental solution of the equation

- y^{″} + V y = E y,

(1.13)

satisfying the initial conditions

φ |_{x = 0} = 0, φ^{'} |_{x = 0} = 1.

The norming constant

h_{s n}

associated with the eigenvalue

m_{n}

is defined by

h_{s n} = 2 π n \log | φ^{'} (1, m_{n}) |, where \log 1 = 0.

(1.14)

Introduce the monodromy matrix

M

for the operator

L

M (λ) = (φ_{j}^{[k - 1]} (1, λ))_{j, k = 1}^{3} .

(1.15)

The matrix-valued function

M

is entire and real on

R

. The characteristic polynomial

D

of the monodromy matrix is given by

D (τ, λ) = det (M (λ) - τ 1 1_{3}), (τ, λ) \in C^{2} .

(1.16)

An eigenvalue of

M

is called a multiplier, it is a zero of the polynomial

D (\cdot, λ)

. The matrix

M

has exactly

3

(counting with multiplicities) eigenvalues

τ_{j}, j = 1, 2, 3

, which satisfy

τ_{1} τ_{2} τ_{3} = 1.

In particular, each

τ_{j} \neq 0

for all

λ \in C

. The multipliers

τ_{1}, τ_{2}, τ_{3}

constitute three distinct branches of some function analytic on a 3-sheeted Riemann surface

R

. This surface is an invariant of the Boussinesq flow. The surface

R

has only algebraic singularities in

C

. These singularities are ramifications of the surface

R

. We consider the ramifications in more detail in our paper Badanin and Korotyaev (To appear).

Define norming constants for the third-order operator. Consider the transpose operator

\tilde{L} \tilde{y} = - ({\tilde{y}}^{″} + p \tilde{y})^{'} - p {\tilde{y}}^{'} + q \tilde{y}, \tilde{y} (0) = \tilde{y} (1) = \tilde{y} (2) = 0.

(1.17)

Theorem 1.1 yields that if

u \in B_{1} (ε)

, then there is exactly one simple eigenvalue

{\tilde{μ}}_{n}

of the operator

\tilde{L}

in each domain

D_{n}, n \in Z_{0}

. Let

{\tilde{y}}_{n} (x)

be a corresponding eigenfunction such that

{\tilde{y}}_{n}^{'} (0) = 1

. There exists a multiplier, which is simple and positive on the half-line

[1, + \infty)

. Denote it by

τ_{3}

. Define the norming constants

h_{s n}, n \in N

, by the identities

h_{s n} = 8 (π n)^{2} \log | {\tilde{y}}_{n}^{'} (1) τ_{3}^{- 1 / 2} ({\tilde{μ}}_{n}) |, τ_{3}^{1 / 2} ({\tilde{μ}}_{n}) > 0.

(1.18)

Consider the McKean transformation, mentioned above, in more detail. This is a standard transformation that lowers the order of the equation so that the third-order equation (1.3) becomes the Hill Equation (1.13) with the energy-dependent potential

V = V (E, u) = E - \frac{p}{2} - \frac{3}{2} \frac{f_{3}^{[2]}}{f_{3}} + \frac{3}{4} {(\frac{f_{3}^{'}}{f_{3}})}^{2}, E = \frac{3}{4} λ^{2 / 3},

where

f_{3}

is the Floquet solution to Equation (1.3) corresponding to the multiplier

τ_{3}

. We consider this transformation in Badanin and Korotyaev (2024c), where we proved the following results:

Theorem 1.2

Let $(u, n) \in B_{1} (ε) \times N$ , let $m_{n}$ be the eigenvalue of the Dirichlet problem for Equation (1.13) with the potential $V = V (E, u)$ , and let $h_{s n}$ be the corresponding norming constant, given by (1.14). Let $h_{s n}$ be given by (1.18). Then

m_{n} = \frac{3}{4} {\tilde{μ}}_{n}^{2 / 3}, h_{s n} = \frac{h_{s n}}{4 π n} .

(1.19)

In this article, we determine asymptotics of the norming constants, given by (1.18).

Theorem 1.3

Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then

| h_{s n} - a_{n} + \sqrt{3} b_{n} | ⩽ \frac{C ‖ u ‖_{1}^{2}}{n},

(1.20)

for all

n \in N

and for some

C > 0

, where

a_{n}

and

b_{n}

are given by (1.12).

Remark

(1)

Using the symmetry (3.26) we can define the norming constants for $n ⩽ - 1$ and extend the estimates (1.20) onto this case, see Corollary 4.8.

(2)

The sequences $h_{c n} = μ_{n} - μ_{n}^{o}$ and $h_{s n}, n \in Z_{0}$ , form the set of spectral data of the inverse problem for the operator $L$ , see Badanin and Korotyaev (2024a).

2. Estimates of the Fundamental Matrix

2.1. The Fundamental Matrix

Introduce the fundamental matrix $Φ (x, λ), (x, λ) \in R \times C$ , of Equation (1.3) by

Φ = (Φ_{j k})_{j, k = 1}^{3} = (\begin{matrix} φ_{1} & φ_{2} & φ_{3} \\ φ_{1}^{'} & φ_{2}^{'} & φ_{3}^{'} \\ φ_{1}^{[2]} & φ_{2}^{[2]} & φ_{3}^{[2]} \end{matrix}), Φ |_{x = 0} = 1 1_{3},

(2.1)

where

φ_{1}, φ_{2}, φ_{3}

are the fundamental solutions,

y^{[2]} = y^{″} + p y

1 1_{3}

is the

3 \times 3

identity matrix. The fundamental matrix

Φ

satisfies the equation

Φ^{'} = H Φ, Φ |_{x = 0} = 1 1_{3}, H = H_{0} - Q,

(2.2)

where the

3 \times 3

matrix-valued functions

H_{0}, Q

have the form

H_{0} = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ λ & 0 & 0 \end{matrix}), Q = (\begin{matrix} 0 & 0 & 0 \\ p s i & 0 & 0 \\ q & p & 0 \end{matrix}) .

(2.3)

The matrix

Φ

satisfies the Liouville identity

(det Φ)^{'} = det Φ Tr (Φ^{'} Φ^{- 1}) = det Φ Tr H = 0,

which yields

det Φ = 1.

The standard arguments, see, e.g. Badanin and Korotyaev (2014a), Lm 2.1, show that each of the matrix-valued function

Φ (x, \cdot), x \in R_{+}

, is entire and real on

R

. Moreover, each of

Φ (x, λ, \cdot), (x, λ) \in R_{+} \times C

is analytic in the Hilbert space

H

given by (1.2).

Consider the unperturbed equation $y^{‴} = λ y$ . In this case $Q = 0$ and the corresponding fundamental matrix $Φ_{0}$ has the form $Φ_{0} = e^{x H_{0}} .$ The eigenvalues of the matrix $H_{0}$ are given by $z, ω z, ω^{2} z$ , here and below

ω = e^{i 2 π / 3} .

Then the eigenvalues of the matrix

Φ_{0}

have the form

e^{z x}, e^{ω z x}, e^{ω^{2} z x}

Consider the transpose operator $\tilde{L}$ , given by (1.17), and the corresponding equation

- ({\tilde{y}}^{″} + p \tilde{y})^{'} - p {\tilde{y}}^{'} + q \tilde{y} = λ \tilde{y} .

(2.4)

Introduce the fundamental solutions

{\tilde{φ}}_{1}, {\tilde{φ}}_{2}, {\tilde{φ}}_{3}

to Equation (2.4) and the fundamental matrix

\tilde{Φ}

\tilde{Φ} = (\begin{matrix} {\tilde{φ}}_{1} & {\tilde{φ}}_{2} & {\tilde{φ}}_{3} \\ {\tilde{φ}}_{1}^{'} & {\tilde{φ}}_{2}^{'} & {\tilde{φ}}_{3}^{'} \\ {\tilde{φ}}_{1}^{[2]} & {\tilde{φ}}_{2}^{[2]} & {\tilde{φ}}_{3}^{[2]} \end{matrix}), \tilde{Φ} |_{x = 0} = 1 1_{3} .

(2.5)

Lemma 2.1

Let $u \in H$ , where $H$ is given by (1.2). Then the matrix-valued functions $Φ$ and $\tilde{Φ}$ satisfy

{\tilde{Φ}}^{⊤} J Φ = J, \tilde{Φ} = J (Φ^{⊤})^{- 1} J, where J = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) .

(2.6)

Proof.

Introduce the matrix coefficient $\tilde{H}$ for the transpose Equation (2.4) by

\tilde{H} = (\begin{matrix} 0 & 1 & 0 \\ - p & 0 & 1 \\ q - λ & - p & 0 \end{matrix}) .

Then

Φ^{'} = H Φ

, see (2.2),

{\tilde{Φ}}^{'} = \tilde{H} \tilde{Φ}

, and

Φ |_{x = 0} = \tilde{Φ} |_{x = 0} = 1 1_{3} .

Moreover, we have

J^{⊤} = J

J^{2} = 1 1_{3}

, and

J H = - (J \tilde{H})^{⊤}

. Using the identities

J Φ^{'} = J H Φ, ({\tilde{Φ}}^{'})^{⊤} J = {\tilde{Φ}}^{⊤} (J \tilde{H})^{⊤} = - {\tilde{Φ}}^{⊤} J H,

we obtain

({\tilde{Φ}}^{⊤} J Φ)^{'} = ({\tilde{Φ}}^{⊤})^{'} J Φ + {\tilde{Φ}}^{⊤} J Φ^{'} = - {\tilde{Φ}}^{⊤} J H Φ + {\tilde{Φ}}^{⊤} J H Φ = 0,

which yields (2.6).

In the unperturbed case $u = 0$ the fundamental solutions have the form

φ_{j}^{0} = \frac{1}{3 z^{j - 1}} \sum_{n = 1}^{3} ω^{n (1 - j)} e^{ω^{n} z x}, {\tilde{φ}}_{j}^{0} = \frac{(- 1)^{j - 1}}{3 z^{j - 1}} \sum_{n = 1}^{3} ω^{n (1 - j)} e^{- ω^{n} z x}, j = 1, 2, 3.

(2.7)

In addition, the following identities hold:

\begin{matrix} φ_{2}^{0} (1, μ_{n}^{o}) = \frac{ξ_{n}^{-} e^{z_{n}^{o}}}{2 \sqrt{3} π n}, {\tilde{φ}}_{2}^{0} (1, μ_{n}^{o}) = \frac{(- 1)^{n} e^{π n / \sqrt{3}} ξ_{n}^{-}}{2 \sqrt{3} π n}, \end{matrix}

(2.8)

\begin{matrix} φ_{3}^{0} (1, μ_{n}^{o}) = \frac{ξ_{n}^{-} e^{z_{n}^{o}}}{(2 π n)^{2}}, {\frac{d φ_{3}^{0} (1, z^{3})}{d z} |}_{z = z_{n}^{o}} = \frac{ξ_{n}^{-} e^{z_{n}^{o}}}{(2 π n)^{2}} (1 - \frac{\sqrt{3}}{π n}), \end{matrix}

(2.9)

\begin{matrix} {\tilde{φ}}_{3}^{0} (1, μ_{n}^{o}) = \frac{(- 1)^{n + 1} e^{π n / \sqrt{3}} ξ_{n}^{-}}{(2 π n)^{2}}, {\frac{d {\tilde{φ}}_{3}^{0} (1, z^{3})}{d z} |}_{z = z_{n}^{o}} = \frac{(- 1)^{n} e^{π n / \sqrt{3}} ξ_{n}^{-}}{(2 π n)^{2}} (1 + \frac{\sqrt{3}}{π n}), \end{matrix}

(2.10)

here and below

ξ_{n}^{\pm} = 1 \pm (- 1)^{n} e^{- \sqrt{3} π n} .

(2.11)

2.2. Transformations of Equation (2.2)

Our method for obtaining asymptotics uses the usual transition from differential equations to integral ones. The asymptotic behavior of the solutions is obtained by iterations in the integral equations. To improve the convergence of the iterations, we will first carry out some transformations of the differential Equation (2.2).

Let $Ψ$ be a solution to Equation (2.2) such that $det Ψ \neq 0$ . Then the fundamental matrix satisfies

Φ (x, λ) = Ψ (x, λ) Ψ^{- 1} (0, λ) .

(2.12)

Let

λ \in C ∖ {0}

. Introduce the matrix

U_{1}

U_{1} = (\begin{matrix} 1 & 1 & 1 \\ ω z & ω^{2} z & z \\ ω^{2} z^{2} & ω z^{2} & z^{2} \end{matrix}), then U_{1}^{- 1} = \frac{1}{3} (\begin{matrix} 1 & \frac{ω^{2}}{z} & \frac{ω}{z^{2}} \\ 1 & \frac{ω}{z} & \frac{ω^{2}}{z^{2}} \\ 1 & \frac{1}{z} & \frac{1}{z^{2}} \end{matrix}), det U_{1} = - i 3 \sqrt{3} λ .

(2.13)

Introduce the matrix-valued function

Y_{1}

by the identity

Y_{1} = U_{1}^{- 1} Ψ .

(2.14)

Lemma 2.2

Let $(λ, u) \in (C ∖ {0}) \times H$ , where $H$ is given by (1.2). Then the matrix-valued function $Y_{1}$ , given by (2.14), satisfies the equation

Y_{1}^{'} - z Ω Y_{1} = \frac{F_{1}}{z} Y_{1},

(2.15)

where

Ω = diag (ω, ω^{2}, 1),

(2.16)

F_{1} = - \frac{p Ω_{1}}{3} - \frac{q Ω_{2}}{3 z}, Ω_{1} = (\begin{matrix} ω^{2} & - ω & - 1 \\ - ω^{2} & ω & - 1 \\ - ω^{2} & - ω & 1 \end{matrix}), Ω_{2} = (\begin{matrix} ω & ω & ω \\ ω^{2} & ω^{2} & ω^{2} \\ 1 & 1 & 1 \end{matrix}) .

(2.17)

Moreover,

(det Y_{1})^{'} = 0.

(2.18)

Proof.

Substituting (2.14) into Equation (2.2) and using the identity

U_{1}^{- 1} (H_{0} - Q) U_{1} = z Ω + \frac{F_{1}}{z},

we obtain Equation (2.15), here

H_{0}, Q

, and

U_{1}

and are given by (2.13) and (2.3). The Liouville identity gives

(det Y_{1})^{'} = det Y_{1} Tr (z Ω + \frac{F_{1}}{z}) = 0,

which yields (2.18).

Equation (2.15) has a diagonal coefficient on the left side and the coefficient, decreasing as $O (\frac{1}{z})$ , on the right side. If $p^{'}, q \in L^{2} (T)$ , then we can get the coefficient, decreasing as $O (1 / z^{2})$ on the right side of the differential equation. Let $u \in H^{1}$ , where $H^{1}$ is given by (1.7). Introduce the domain $Λ$ in $C$ by

Λ = {λ \in C : | λ | > 1} .

(2.19)

Introduce the matrix-valued function

U_{2}

U_{2} = 1 1_{3} + \frac{p Ω_{3}}{3 z^{2}}, det U_{2} = 1,

(2.20)

where

Ω_{3} = \frac{i}{\sqrt{3}} (\begin{matrix} 0 & ω & - ω \\ - ω^{2} & 0 & ω^{2} \\ 1 & - 1 & 0 \end{matrix}) .

(2.21)

Introduce the matrix

Y_{2}

by the identity

Y_{2} = U_{2}^{- 1} Y_{1} .

(2.22)

Introduce the matrix-valued function

L = (\begin{matrix} 0 & - ω v & - ω \bar{v} \\ - ω^{2} \bar{v} & 0 & - ω^{2} v \\ - v & - \bar{v} & 0 \end{matrix}), where v = \frac{i p^{'}}{\sqrt{3}} + q .

(2.23)

In the following lemma we rewrite Equation (2.15) in the form (2.24) having a diagonal coefficient on the left side and the coefficient, decreasing as

O (\frac{1}{z^{2}})

on the right side. Below we use the standard operator norm of matrices

| A | = sup_{| h | = 1} | A h |, | h |^{2} = \sum_{j = 1}^{3} | h_{j} |^{2}

Lemma 2.3

Let $(λ, u) \in Λ \times H^{1}$ . Then the function $Y_{2}$ satisfies the equation

Y_{2}^{'} - z Θ Y_{2} = \frac{F_{2}}{z^{2}} Y_{2},

(2.24)

where

Θ = Ω - \frac{p Ω^{2}}{3 z^{2}} - \frac{q Ω}{3 z^{3}} - \frac{p p^{'} Ω^{2}}{27 z^{5}}, F_{2} = z^{2} (U_{2}^{- 1} (z Ω U_{2} + \frac{F_{1} U_{2}}{z} - U_{2}^{'}) - z Θ),

(2.25)

F_{1}

is given by (2.17). Moreover, the diagonal entries of

F_{2}

vanish:

F_{2, j j} = 0, j = 1, 2, 3,

(2.26)

and

F_{2}

satisfies the estimate

\begin{aligned} sup_{x \in R} | F_{2} (x, λ) - \frac{L}{3} | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |}, \end{aligned}

(2.27)

for some

C > 0

, where the matrix

L

is given by (2.23), the norm

‖ u ‖_{1}

is defined by (1.8). Moreover,

(det Y_{2})^{'} = 0.

(2.28)

Proof.

Substituting the identity (2.22) into Equation (2.15) we obtain

Y_{2}^{'} = A Y_{2},

(2.29)

where

A = U_{2}^{- 1} (B U_{2} - U_{2}^{'}), B = z Ω + \frac{F_{1}}{z} = z Ω - \frac{p Ω_{1}}{3 z} - \frac{q Ω_{2}}{3 z^{2}},

(2.30)

here we used (2.17). The diagonal entries of the matrix

A

has the form

A_{j j} = z Ω_{j j} + \frac{F_{1, j j}}{z} - (U_{2}^{- 1} U_{2}^{'})_{j j}, j = 1, 2, 3.

(2.31)

The definitions (2.16) and (2.17) imply

Ω_{j j} = ω^{j}, F_{1, j j} = - \frac{ω^{2 j} p}{3} - \frac{ω^{j} q}{3 z}, j = 1, 2, 3.

(2.32)

The definition (2.21) gives

Ω_{3}^{2} = - \frac{1}{3} (\begin{matrix} ω^{2} & ω & 1 \\ ω^{2} & ω & 1 \\ ω^{2} & ω & 1 \end{matrix}), Ω_{3}^{3} = 0,

(2.33)

then the definition (2.20) implies

U_{2}^{- 1} = 1 1_{3} - \frac{p Ω_{3}}{3 z^{2}} + \frac{p^{2} Ω_{3}^{2}}{9 z^{4}} .

(2.34)

The relations (2.20) and (2.34) yield

U_{2}^{- 1} U_{2}^{'} = \frac{p^{'} Ω_{3}}{3 z^{2}} - \frac{p p^{'} Ω_{3}^{2}}{9 z^{4}} .

(2.35)

The relations (2.21) and (2.33) give

(U_{2}^{- 1} U_{2}^{'})_{j j} = \frac{ω^{2 j} p p^{'}}{27 z^{4}}, j = 1, 2, 3.

(2.36)

Substituting the identities (2.32) and (2.36) into (2.31) we obtain

A_{j j} = z ω^{j} - \frac{ω^{2 j} p}{3 z} - \frac{ω^{j} q}{3 z^{2}} - \frac{ω^{2 j} p p^{'}}{27 z^{4}} = z Θ_{j j}, j = 1, 2, 3.

The definitions (2.25) and (2.30) give

F_{2} = z^{2} (A - z Θ) .

(2.37)

This yields (2.26). Moreover, Equation (2.29) takes the form (2.24).

We prove (2.27). The definition (2.20) and the identity (2.34) give

U_{2}^{- 1} B U_{2} = (1 1_{3} - \frac{p Ω_{3}}{3 z^{2}} + \frac{p^{2} Ω_{3}^{2}}{9 z^{4}}) B (1 1_{3} + \frac{p Ω_{3}}{3 z^{2}}) = B + \frac{p}{3 z^{2}} (B Ω_{3} - Ω_{3} B) + \frac{α_{1}}{z^{4}},

where

sup_{x \in R} | α_{1} (x, λ) | ⩽ C ‖ u ‖^{2}

, for some

C > 0

. Substituting the definition (2.30) into this identity we obtain

U_{2}^{- 1} B U_{2} = z Ω + \frac{A_{1}}{z} - \frac{q Ω_{2}}{3 z^{2}} + \frac{α_{2}}{z^{3}},

where

sup_{x \in R} | α_{2} (x, λ) | ⩽ C ‖ u ‖^{2}

, for some

C > 0

A_{1} = \frac{p}{3} (Ω Ω_{3} - Ω_{3} Ω - Ω_{1}) .

(2.38)

Substituting this identity and (2.35) into the definition (2.30) we obtain

A = U_{2}^{- 1} B U_{2} - U_{2}^{- 1} U_{2}^{'} = z Ω + \frac{A_{1}}{z} + \frac{A_{2}}{z^{2}} + \frac{α_{3}}{z^{3}},

(2.39)

where

sup_{x \in R} | α_{3} (x, λ) | ⩽ C ‖ u ‖_{1}^{2}

, for some

C > 0

A_{2} = - \frac{1}{3} (q Ω_{2} + p^{'} Ω_{3}) .

(2.40)

The definitions (2.16), (2.17), (2.21), and (2.23) give

Ω Ω_{3} - Ω_{3} Ω - Ω_{1} + Ω^{2} = 0, L = q (Ω - Ω_{2}) - p^{'} Ω_{3} .

Substituting these identities into (2.38) and (2.40) we obtain

A_{1} = - \frac{p}{3} Ω^{2}, A_{2} = \frac{L}{3} - \frac{q}{3} Ω .

Then the relations (2.25) and (2.39) imply

A = z Ω - \frac{p Ω^{2}}{3 z} - \frac{q Ω}{3 z^{2}} + \frac{L}{3 z^{2}} + \frac{α_{4}}{z^{3}} = z Θ + \frac{L}{3 z^{2}} + \frac{α_{5}}{z^{3}},

(2.41)

where

sup_{x \in R} | α_{j} (x, λ) | ⩽ C ‖ u ‖_{1}^{2}, j = 4, 5

, for some

C > 0

Θ

is given by (2.25). Then the identities (2.37) and (2.41) yield

F_{2} = \frac{L}{3} + \frac{α_{5}}{z} .

This gives (2.27). The Liouville identity gives

(det Y_{2})^{'} = det Y_{2} Tr (z Θ + \frac{F_{2}}{z^{2}}) = 0,

which yields (2.28).

2.3. Representation of the Fundamental Matrix

Fundamental solutions $φ_{j}$ are not very convenient for asymptotic analysis. Each of them contains both exponentially increasing and exponentially decreasing at high energies terms. The contribution to the asymptotics from decreasing terms is lost against the background of increasing ones. Below, in Lemmas 2.4 and 2.5 (ii), we express the fundamental matrix in terms of other, more appropriate, fundamental solutions $ϕ_{1, j}$ and $ϕ_{2, j}$ , given by (3.1). The method we will use goes back to Birkhoff, see Naimark (1967). We formulate our results for $λ \in C_{+}$ . In this case we have the simple estimates

ℜ (ω z) ⩽ ℜ (ω^{2} z) ⩽ ℜ z, \forall λ \in C_{+} .

(2.42)

The analytic extension gives the corresponding results in

C_{-}

Let $λ \in Λ_{+}$ , where $Λ_{+}$ is the domain in $C_{+}$ given by

Λ_{+} = {λ \in C_{+} : | λ | > 1} .

(2.43)

Let

N \in N

. Consider the Birkhoff integral equation

X_{1} = 1 1_{3} + \frac{K_{1} X_{1}}{z},

(2.44)

on the space

C ([0, N])

3 \times 3

matrix-valued functions continuous in

x

, where

\begin{matrix} (K_{1} X_{1})_{l j} (x, λ) = \int_{0}^{N} K_{1, l j} (x, s, λ) (F_{1} X_{1})_{l j} (s, λ) d s, l, j = 1, 2, 3, \end{matrix}

(2.45)

\begin{matrix} K_{1, l j} (x, s, λ) = {\begin{cases} e^{z (x - s) (ω^{l} - ω^{j})} χ (x - s), & l < j \\ - e^{z (x - s) (ω^{l} - ω^{j})} χ (s - x), & l ⩾ j \end{cases}, χ (s) = {\begin{cases} 1, & s ⩾ 0 \\ 0, & s < 0 \end{cases}, \end{matrix}

(2.46)

F_{1}

is given by (2.17). Using the estimates (2.42) we obtain

| K_{1, l j} (x, s, λ) | ⩽ 1

for all

l, j = 1, 2, 3

(x, s, λ) \in [0, N]^{2} \times C_{+}

Introduce the balls $B (r), r > 0$ , in the space $H$ , having the form (1.2), by

B (r) = {u \in H : ‖ u ‖ < r}, ‖ u ‖^{2} = ‖ p ‖^{2} + ‖ q ‖^{2} .

We prove the following results about the integral Equation (2.44).

Lemma 2.4

Let $N \in N$ . If $(λ, u) \in Λ_{+} \times B (1 / 6 N)$ , then Equation (2.44) has a unique solution $X_{1} \in C ([0, N])$ . Each matrix-valued function $X_{1} (\cdot, λ)$ , $λ \in Λ_{+}$ , is continuous on $[0, N]$ , each $X_{1} (x, \cdot)$ , $x \in [0, N]$ , is analytic on the domain $Λ_{+}$ , and satisfies

X_{1, l j} (0, λ) = X_{1, j l} (N, λ) = 0, 1 ⩽ l < j ⩽ 3, X_{1, j j} (N, λ) = 1, j = 1, 2, 3,

(2.47)

sup_{x \in [0, N]} | X_{1} (x, λ) | ⩽ 2, sup_{x \in [0, N]} | X_{1} (x, λ) - 1 1_{3} | ⩽ \frac{6 N ‖ u ‖}{| z |},

(2.48)

for all

λ \in Λ_{+}

. The function

Y_{1} = X_{1} e^{z x Ω}

(2.49)

is the solution to Equation (2.15) on the interval

[0, N]

, satisfying the conditions

Y_{1, l j} (0, λ) = Y_{1, j l} (N, λ) = 0, 1 ⩽ l < j ⩽ 3, Y_{1, j j} (N, λ) = e^{N ω^{j} z}, j = 1, 2, 3,

(2.50)

here

Ω

is given by (2.16). The function

Ψ_{1} = (Ψ_{1, l j})_{l, j = 1}^{3} = U_{1} Y_{1} = U_{1} X_{1} e^{z x Ω}

(2.51)

satisfies Equation (2.2) on the interval

[0, N]

, here

U_{1}

is given by (2.13). Moreover,

det X_{1} = det Y_{1} = 1, det Ψ_{1} = - i 3 \sqrt{3} λ .

(2.52)

Each of the functions

X_{1}^{- 1} (\cdot, λ)

λ \in Λ_{+}

, is continuous on

[0, N]

, each

X_{1}^{- 1} (x, \cdot)

x \in [0, N]

, is analytic on the domain

Λ_{+}

. The fundamental matrix satisfies

Φ (x, λ) = Ψ_{1} (x, λ) Ψ_{1}^{- 1} (0, λ) = U_{1} (λ) X_{1} (x, λ) e^{z x Ω} X_{1}^{- 1} (0, λ) U_{1}^{- 1} (λ),

(2.53)

for all

x \in [0, N]

. Each of the functions

X_{1} (x, λ, \cdot)

X_{1}^{- 1} (x, λ, \cdot)

(x, λ) \in [0, N] \times Λ_{+}

, is analytic on the ball

B (1 / 6 N)

. If, in addition,

u \in B (1 / 12 N)

, then

sup_{x \in [0, N]} | X_{1}^{- 1} (x, λ) - 1 1_{3} | ⩽ \frac{12 N ‖ u ‖}{| z |} .

(2.54)

Proof.

Iterations in the integral Equation (2.44) give

X_{1} = \sum_{n = 0}^{\infty} \frac{K_{1}^{n} 1 1_{3}}{z^{n}} .

(2.55)

Moreover, the definition (2.45) gives the estimate

sup_{x \in [0, N]} | (K_{1} X) (x, λ) | ⩽ \int_{0}^{N} | F_{1} (s, λ) | | X (s, λ) | d s,

for a continuous

3 \times 3

matrix-valued function

X

and the definitions (2.17) imply

\int_{0}^{1} | F_{1} (s, λ) | d s ⩽ 3 (‖ p ‖ + \frac{‖ q ‖}{| z |}) ⩽ 3 ‖ u ‖,

| z | ⩾ 1

. Then

sup_{x \in [0, N]} | K_{1}^{n} 1 1_{3} (x, λ) | ⩽ {(\int_{0}^{N} | F_{1} (s) | d s)}^{n} ⩽ (3 N ‖ u ‖)^{n}, n ⩾ 0.

Therefore, the series (2.55) converges absolutely and uniformly on any bounded subset of

[0, N] \times Λ_{+} \times B (\frac{1}{3 N})

. Each term is a continuous function of

x \in [0, N]

and an analytic function of

λ \in C

and

u \in H

, then the sum is continuous in

x \in [0, N]

and analytic in

λ \in Λ_{+}

and

u \in B (\frac{1}{3 N})

. Summing the majorants in the series (2.55) we obtain

sup_{x \in [0, N]} | X_{1} (x, λ) | ⩽ \frac{1}{1 - A}, sup_{x \in [0, N]} | X_{1} (x, λ) - 1 1_{3} | ⩽ \frac{A}{1 - A}, A = \frac{3 N ‖ u ‖}{| z |} .

(2.56)

u \in B (\frac{1}{6 N})

, then

A ⩽ \frac{1}{2}

, which yields (2.48). The definitions (2.45) and (2.46) give

(K_{1} X_{1})_{l j} |_{x = 0} = 0

l < j

(K_{1} X_{1})_{l j} |_{x = N} = 0

l ⩾ j

. This gives (2.47).

In Badanin and Korotyaev (2021), Cor 5.1 we proved that $Y_{1}$ , given by (2.49), is the solution to Equation (2.15) on the interval $[0, N]$ . The identities (2.47) provide (2.50). The identity (2.14) implies (2.51). The identities (2.18) and (2.50) imply $det Y_{1} = det e^{N ω^{j} z} = 1$ . Then the identities (2.13) and (2.49) give (2.52). Therefore, $X_{1}^{- 1}$ exists and is continuous with respect to $x$ and analytic with respect to $λ$ and $u$ . The relations (2.12) and (2.51) give (2.53).

The estimate (2.56) shows that if $u \in B (\frac{1}{12 N})$ , then

sup_{x \in [0, N]} | X_{1} (x, λ) | ⩾ 1 - sup_{x \in [0, N]} | X_{1} (x, λ) - 1 1_{3} | ⩾ 1 - 2 A ⩾ \frac{1}{2},

and

\begin{aligned} sup_{x \in [0, N]} | X_{1}^{- 1} (x, λ) - 1 1_{3} | = sup_{x \in [0, N]} | X_{1}^{- 1} (x, λ) | | 1 1_{3} - X_{1} (x, λ) | ⩽ 2 sup_{x \in [0, N]} | 1 1_{3} - X_{1} (x, λ) | ⩽ 4 A . \end{aligned}

The definition of

A

in (2.56) yields (2.54).

2.4. Representation for the Case

p^{'}, q \in L^{1} (T)

Let $u \in H^{1}$ , where $H^{1}$ is given by (1.7), let $N \in N$ , and let $K_{2}$ be an integral operator on the space $C [0, N]$ of $3 \times 3$ continuous matrix-valued functions on $[0, N]$ given by

(K_{2} X_{2})_{l j} (x, λ) = \int_{0}^{N} K_{2, l j} (x, s, λ) (F_{2} X_{2})_{l j} (s, λ) d s \forall l, j = 1, 2, 3,

(2.57)

where

K_{2, l j} (x, s, λ) = {\begin{cases} e^{z \int_{s}^{x} (Θ_{l} (u, λ) - Θ_{j} (u, λ)) d u} χ (x - s), & l < j \\ - e^{- z \int_{x}^{s} (Θ_{l} (u, λ) - Θ_{j} (u, λ)) d u} χ (s - x), & l ⩾ j \end{cases}, χ (s) = {\begin{cases} 1, & s ⩾ 0 \\ 0, & s < 0 \end{cases},

(2.58)

F_{2}

and

Θ

are given by (2.25). Consider the integral equation

X_{2} = 1 1_{3} + \frac{K_{2} X_{2}}{z^{2}}

(2.59)

on the space

C [0, N]

. Introduce the matrix

N = (N_{l j})_{l, j = 1}^{3}, N_{l j} (x, t, λ) = K_{1, l j} (x, t, λ) L_{l j} (t) .

(2.60)

where

K_{1, l j}

have the form (2.46),

L

is given by (2.23). We prove the following results.

Lemma 2.5

Let $N \in N$ and let $ε > 0$ be small enough. (i)

If $(λ, u) \in Λ_{+} \times B_{1} (ε)$ , then the integral Equation (2.59) has a unique solution $X_{2} \in C ([0, N])$ . Each matrix-valued function $X_{2} (\cdot, λ)$ , $λ \in Λ_{+}$ , is continuous on $[0, N]$ , each $X_{2} (x, \cdot)$ , $x \in [0, N]$ , is analytic on the domain $Λ_{+}$ , and

X_{2, l j} (0, λ) = X_{2, j l} (N, λ) = 0, 0 ⩽ l < j ⩽ 3, X_{2, j j} (N, λ) = 1, j = 1, 2, 3.

(2.61)

Moreover, each matrix-valued function

X_{2} (x, λ, \cdot), (x, λ) \in [0, N] \times Λ_{+}

, is analytic on

B_{1} (ε)

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ . Then the matrix $X_{2}$ satisfies

sup_{x \in [0, N]} | X_{2} (x, λ) | ⩽ C, sup_{x \in [0, N]} | X_{2} (x, λ) - \sum_{n = 0}^{l} \frac{R_{n} (x, λ)}{z^{2 n}} | ⩽ \frac{C ‖ u ‖_{1}^{l + 1}}{| z |^{2 (l + 1)}}, l = 0, 1, 2, \dots,

(2.62)

for some

C > 0

, where

R_{n} = K_{2}^{n} 1 1_{3} .

(2.63)

Each

X_{2}^{- 1} (\cdot, λ)

λ \in Λ_{+}

, is continuous on

[0, N]

, each

X_{2}^{- 1} (x, \cdot)

x \in [0, N]

, is analytic on the domain

Λ_{+}

. Furthermore,

sup_{x \in [0, N]} | R_{1} (x, λ) - \frac{1}{3} \int_{0}^{N} N (x, s, λ) d s | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |},

(2.64)

for some

C > 0

, where

N

is given by (2.60).

(ii)

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ . Then the function

Y_{2} (x, λ) = X_{2} (x, λ) e^{z \int_{0}^{x} Θ (s, λ) d s}

(2.65)

is the solution to Equation (2.24) on the interval

[0, N]

, satisfying the conditions

Y_{2, l j} (0, λ) = Y_{2, j l} (N, λ) = 0, 1 ⩽ l < j ⩽ 3, Y_{2, j j} (N, λ) = e^{z \int_{0}^{N} Θ_{j} (s, λ) d s}, j = 1, 2, 3.

(2.66)

The matrix-valued function

Ψ_{2} (x, λ) = (Ψ_{2, l j} (x, λ))_{l, j = 1}^{3} = U_{1} (λ) U_{2} (x, λ) X_{2} (x, λ) e^{z \int_{0}^{x} Θ (s, λ) d s}, x \in [0, N],

(2.67)

satisfies Equation (2.2), here

U_{1}

is given by (2.13),

U_{2}

has the form (2.20). Moreover,

det X_{2} = det Y_{2} = 1, det Ψ_{2} = - i 3 \sqrt{3} λ .

(2.68)

The fundamental matrix satisfies

Φ (x, λ) = Ψ_{2} (x, λ) Ψ_{2}^{- 1} (0, λ) = U_{1} (λ) U_{2} (x, λ) X_{2} (x, λ) e^{z \int_{0}^{x} Θ (s, λ) d s} X_{2}^{- 1} (0, λ) U_{2}^{- 1} (0, λ) U_{1}^{- 1} (λ),

(2.69)

for all

x \in [0, N]

Proof.

(i) Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ for some $ε > 0$ small enough. The definition of $Θ$ in (2.25) yields that the kernel $K_{2, l j} (x, s, λ)$ of the integral operator $K_{2}$ , given by (2.57), is bounded, then

sup_{x \in [0, N]} | K_{2} X_{2} (x, λ) | ⩽ C ‖ u ‖_{1} \int_{0}^{N} | X_{2} (x, λ) | d x,

for some

C > 0

. Repeating the arguments from the proof of Lemma 2.4 we obtain that Equation (2.59) has a unique solution

X_{2} \in C ([0, N])

. The definitions (2.57) and (2.58) yield

(K_{2} X_{2})_{l j} |_{x = 0} = 0, l < j

(K_{2} X_{2})_{l j} |_{x = N} = 0, l ⩾ j

. This gives (2.61). Iterations in Equation (2.59) give

X_{2} = \sum_{n = 0}^{\infty} \frac{R_{n}}{z^{2 n}},

the series converges absolutely and uniformly on any bounded subset of

[0, N] \times Λ_{+} \times B_{1} (ε)

and the estimates (2.62) hold true. Each term is continuous in

x \in [0, N]

and analytic in

λ \in Λ_{+}

and

u \in B_{1} (ε)

, then the sum is also continuous in

x

and analytic in

λ

and

u

Let $l, j = 1, 2, 3$ . Then the definitions (2.57), (2.63), and the estimate (2.27) imply

R_{1, l j} (x, λ) = \frac{1}{3} \int_{0}^{N} K_{2, l j} (x, s, λ) L_{l j} (s) d s + \frac{α_{1} (x, λ)}{z}, sup_{x \in [0, N]} | α_{1} (x, λ) | ⩽ C ‖ u ‖_{1}^{2},

(2.70)

for some

C > 0

, where

L

is given by (2.23). The definition (2.25) implies

Θ_{l} - Θ_{j} = ω^{l} - ω^{j} + \frac{α_{2}}{z^{2}}, sup_{x \in [0, N]} | α_{2} (x, λ) | ⩽ C ‖ u ‖ .

Then the definitions (2.46) and (2.58) yield

K_{2, l j} = K_{1, l j} + \frac{α_{3}}{z},

where

sup_{(x, s) \in [0, N]^{2}} | α_{3} (x, s, λ) | ⩽ C ‖ u ‖

. Substituting this asymptotics into (2.70) and using (2.60) we obtain (2.64).

(ii) In Badanin and Korotyaev (2021), Cor 6.2 we proved that $Y_{2}$ , given by (2.65), satisfy Equation (2.24) on the interval $[0, N]$ . The identities (2.61) provide (2.66). The definitions (2.14) and (2.22) imply (2.67). The identities (2.28) and (2.66) yields $det Y_{2} = det e^{z \int_{0}^{N} Θ_{j} (s, λ) d s} = 1$ . Then the identities (2.13), (2.20), and (2.65) give (2.68). The relations (2.12) and (2.67) imply (2.69).

3. 3-Point Eigenvalues

3.1. Two Representations of the Characteristic Function $Δ$

Assume $N = 2$ in Lemmas 2.4 and 2.5. Introduce the fundamental solutions $ϕ_{ν, j} (x, λ)$ , $ν = 1, 2$ , $j = 1, 2, 3$ , $(x, λ) \in [0, 2] \times Λ_{+}$ , to Equation (1.3) by

\begin{aligned} ϕ_{ν, j} = Ψ_{ν, 1 j}, \end{aligned}

(3.1)

where the matrix-valued function

Ψ_{1}

is given by (2.51),

Ψ_{2}

is given by (2.67). Each function

X_{ν} (x, \cdot), ν = 1, 2, x \in [0, 2]

, from Lemmas 2.4 and 2.5, is analytic on

Λ_{+}

, then

ϕ_{ν, j} (x, \cdot), j = 1, 2, 3

, are also analytic.

Introduce the matrix-valued functions $ϕ_{ν}, ν = 1, 2$ , by

ϕ_{ν} (λ) = (\begin{matrix} ϕ_{ν, 1} (0, λ) & ϕ_{ν, 2} (0, λ) & ϕ_{ν, 3} (0, λ) \\ ϕ_{ν, 1} (1, λ) & ϕ_{ν, 2} (1, λ) & ϕ_{ν, 3} (1, λ) \\ ϕ_{ν, 1} (2, λ) & ϕ_{ν, 2} (2, λ) & ϕ_{ν, 3} (2, λ) \end{matrix}), λ \in Λ_{+},

(3.2)

where

ϕ_{ν, k}

are given by (3.1).

Lemma 3.1

Assume that $u \in B (ε)$ and $ν = 1$ or $u \in B_{1} (ε)$ and $ν = 2$ . Then the function $det ϕ_{ν}$ is entire and satisfies

det ϕ_{ν} = - i 3 \sqrt{3} λ Δ,

(3.3)

where the entire function

Δ

is given by (1.5).

Proof.

We have the following simple identity

det ϕ_{ν} (λ) = det Ψ_{ν} (0, λ) Δ (λ), ν = 1, 2,

see, e.g. Badanin and Korotyaev (2021), Lm 3.2. Then the identities (2.52) and (2.68) give (3.3).

In the unperturbed case $p = q = 0$ we have $ϕ_{1} = ϕ_{2} = ϕ_{0}$ , where

ϕ_{0} = (\begin{matrix} 1 & 1 & 1 \\ e^{ω z} & e^{ω^{2} z} & e^{z} \\ e^{2 ω z} & e^{2 ω^{2} z} & e^{2 z} \end{matrix}) .

(3.4)

The function

Δ

has the form

Δ_{0} = \frac{8}{3 \sqrt{3} λ} \sin \frac{\sqrt{3} z}{2} \sin \frac{\sqrt{3} ω z}{2} \sin \frac{\sqrt{3} ω^{2} z}{2} = \frac{4}{3 \sqrt{3} λ} \sin (\frac{\sqrt{3} z}{2}) (\cosh (\frac{3 z}{2}) - \cos (\frac{\sqrt{3} z}{2})) .

(3.5)

3.2. Asymptotics of the Fundamental Solutions

ϕ_{ν}

Below we will use the following simple identities

\begin{aligned} ω + 2 ω^{2} = - ω - 2 = ω^{2} - 1 = i \sqrt{3} ω, ω^{2} + 2 = - ω^{2} - 2 ω = 1 - ω = i \sqrt{3} ω^{2}, \\ 1 + 2 ω = - 1 - 2 ω^{2} = ω - ω^{2} = i \sqrt{3} . \end{aligned}

(3.6)

Introduce the functions

ζ_{j} (x, λ) = \sum_{l = 1}^{3} \int_{0}^{2} N_{l j} (x, t, λ) d t, j = 1, 2, 3,

(3.7)

where the matrix

N

is given by (2.60). Recall that

| K_{1, l j} (x, t, λ) | ⩽ 1

, then the definition (2.60) gives

sup_{(x, λ) \in [0, 2] \times C_{+}} | ζ_{j} (x, λ) | ⩽ C,

(3.8)

for some

C > 0

and for all

j = 1, 2, 3

. Substituting the definitions (2.23) and (2.46) into the definition (2.60) and using the identities (3.6) we obtain

N = (\begin{matrix} 0 & - ω e^{i \sqrt{3} z (x - t)} χ (x - t) v (t) & - ω e^{- i \sqrt{3} ω^{2} z (x - t)} χ (x - t) \bar{v} (t) \\ ω^{2} e^{- i \sqrt{3} z (x - t)} χ (t - x) \bar{v} (t) & 0 & - ω^{2} e^{i \sqrt{3} ω z (x - t)} χ (x - t) v (t) \\ e^{i \sqrt{3} ω^{2} z (x - t)} χ (t - x) v (t) & e^{- i \sqrt{3} ω z (x - t)} χ (t - x) \bar{v} (t) & 0 \end{matrix}),

(3.9)

where

v

is given by (2.23). The identities (3.9) and the definitions (3.7) give

\begin{aligned} ζ_{1} (x, λ) & = \int_{x}^{2} e^{i \sqrt{3} ω^{2} z (x - s)} v (s) d s + ω^{2} \int_{x}^{2} e^{- i \sqrt{3} z (x - s)} \bar{v} (s) d s, \\ ζ_{2} (x, λ) & = - ω \int_{0}^{x} e^{i \sqrt{3} z (x - s)} v (s) d s + \int_{x}^{2} e^{- i \sqrt{3} ω z (x - s)} \bar{v} (s) d s, \\ ζ_{3} (x, λ) & = - ω^{2} \int_{0}^{x} e^{i \sqrt{3} ω z (x - s)} v (s) d s - ω \int_{0}^{x} e^{- i \sqrt{3} ω^{2} z (x - s)} \bar{v} (s) d s . \end{aligned}

(3.10)

We determine asymptotics of

ϕ_{ν, j}

Lemma 3.2

(i)

Let $(λ, u) \in Λ_{+} \times B (\frac{1}{12})$ . Then the fundamental solutions $ϕ_{1, j}$ satisfy

sup_{x \in [0, 2]} | e^{- z x ω^{j}} ϕ_{1, j} (x, λ) - 1 | ⩽ \frac{36 ‖ u ‖}{| z |}, j = 1, 2, 3.

(3.11)

(ii)

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let $j = 1, 2, 3$ . Then the solutions $ϕ_{2, j}$ satisfy

ϕ_{2, j} (x, λ) = (1 + \frac{ω^{j} p (x)}{3 z^{2}} + \frac{w_{j} (x, λ)}{3 z^{2}}) e^{z \int_{0}^{x} Θ_{j} (s, λ) d s}, x \in [0, 2],

(3.12)

where

sup_{x \in [0, 2]} | w_{j} (x, λ) - ζ_{j} (x, λ) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |},

(3.13)

for some

C > 0

Proof.

(i) Let $(λ, u) \in Λ_{+} \times B (\frac{1}{3})$ . The definition (2.13) gives $U_{1, 1 j} = 1, j = 1, 2, 3$ . The definitions (2.51) and (3.1) give

ϕ_{1, j} = e^{z x ω^{j}} \sum_{k = 1}^{3} U_{1, 1 k} X_{1, k j} = e^{z x ω^{j}} \sum_{k = 1}^{3} X_{1, k j}, j = 1, 2, 3.

(3.14)

The relations (2.48) and (3.14) give

ϕ_{1, j} = e^{z x ω^{j}} \sum_{k = 1}^{3} (δ_{k j} + α_{k j}) = e^{z x ω^{j}} (1 + α_{j}), | α_{k j} (x, λ) | ⩽ \frac{12 ‖ u ‖}{| z |}, α_{j} = \sum_{k = 1}^{3} α_{k j}, j, k = 1, 2, 3.

(3.15)

The relations (3.15) give (3.11).

(ii) Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ . Then the identity (2.67) gives

ϕ_{2, j} (x, λ) = \sum_{l = 1}^{3} (U_{2} (x, λ) X_{2} (x, λ))_{l j} e^{z \int_{0}^{x} Θ_{j} (s, λ) d s}, x \in [0, 2],

(3.16)

where

U_{2}

is given by (2.20). The definition (2.20) and the estimate (2.62) imply

sup_{x \in [0, 2]} | U_{2} (x, λ) X_{2} (x, λ) - 1 1_{3} - \frac{1}{z^{2}} (\frac{p (x)}{3} Ω_{3} + R_{1} (x, λ)) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{4}},

for some

C > 0

. The estimates (2.64) give

sup_{x \in [0, 2]} | \sum_{l = 1}^{3} (U_{2} (x, λ) X_{2} (x, λ))_{l j} - 1 - \frac{1}{3 z^{2}} (p (x) \sum_{l = 1}^{3} Ω_{3, l j} + \sum_{l = 1}^{3} \int_{0}^{2} N_{l j} (x, s, λ) d s) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{3}},

(3.17)

for some

C > 0

, where the matrix

N

is given by (2.60). The definition (2.21) and the identities (3.6) imply

\sum_{l = 1}^{3} Ω_{3, l j} = ω^{j}, j = 1, 2, 3.

The definitions (3.7) and the relations (3.16), (3.17) give (3.12) and (3.13).

3.3. Counting Results

Introduce the domain

D = Λ_{+} ∖ \cup_{n \in N} \bar{D_{- n}} .

(3.18)

Recall the standard estimate from Pöschel and Trubowitz (1987), Lm 2.1:

| \sin z | > \frac{1}{4} e^{| ℑ z |}, as | z - π n | ⩾ \frac{π}{4} \forall n \in Z .

(3.19)

We prove the following results.

Lemma 3.3

(i)

The following estimates hold true

\begin{matrix} | e^{(ω - 1) z} | ⩽ e^{- \frac{3}{2}}, \forall λ \in Λ_{+}, \end{matrix}

(3.20)

\begin{matrix} | e^{(ω^{2} - 1) z} - 1 | > \frac{1}{3}, \forall λ \in D . \end{matrix}

(3.21)

(ii)

The function $Δ_{0}$ , given by (3.5), satisfies

| Δ_{0} (λ) | > \frac{| e^{z (ω^{2} + 2)} |}{24 \sqrt{3} | λ |}, λ \in C_{+} ∖ \cup_{n \in Z} D_{n} .

(3.22)

(iii)

Let $u \in B (ε)$ . Then

| \frac{Δ (λ)}{Δ_{0} (λ)} - 1 | ⩽ \frac{C ‖ u ‖}{| z |},

(3.23)

for all

λ \in C_{+} ∖ \cup_{n \in Z} D_{n}

and for some

C > 0

Proof.

(i) If $λ \in Λ_{+}$ , then $| z | ⩾ 1$ and (3.6) implies

| e^{(ω - 1) z} | = | e^{- i \sqrt{3} ω^{2} z} | = e^{- ℜ i \sqrt{3} ω^{2} z} = e^{- \frac{3 x + \sqrt{3} y}{2}} = e^{- \sqrt{3} | z | \sin (φ + \frac{π}{3})} ⩽ e^{- \sqrt{3} \sin (φ + \frac{π}{3})} ⩽ e^{- \frac{3}{2}},

where

z = x + i y = | z | e^{i φ}, 0 ⩽ φ ⩽ \frac{π}{3}

. This yields (3.20).

The definitions (2.16) and the identities (3.6) imply $ω^{2} - 1 = i \sqrt{3} ω = e^{- i \frac{2 π}{3}} - 1 = - \sqrt{3} e^{i \frac{π}{6}}$ , then

| e^{(ω^{2} - 1) z} - 1 | = | e^{i \sqrt{3} ω z} - 1 | = 2 | e^{i \frac{\sqrt{3}}{2} ω z} | | \sinh (i \frac{\sqrt{3}}{2} ω z) | = 2 | e^{- \frac{\sqrt{3}}{2} e^{i \frac{π}{6}} z} | | \sinh (\frac{\sqrt{3}}{2} e^{i \frac{π}{6}} z) | .

Using the estimate

| e^{- \frac{\sqrt{3}}{2} e^{i \frac{π}{6}} z} | = e^{- \frac{\sqrt{3}}{2} ℜ (e^{i \frac{π}{6}} z)} = e^{- \frac{\sqrt{3}}{4} (x \sqrt{3} - y)} ⩾ e^{- \frac{3}{4} x} ⩾ e^{- \frac{3}{8}} > \frac{2}{3},

we obtain

| e^{(ω^{2} - 1) z} - 1 | ⩾ \frac{4}{3} | \sinh (\frac{\sqrt{3}}{2} e^{i \frac{π}{6}} z) | .

Let, in addition,

λ \in D

. Then the estimate (3.19) implies

| \sinh (\frac{\sqrt{3}}{2} e^{i \frac{π}{6}} z) | > \frac{1}{4}

, which yields (3.21).

(ii) The estimate (3.19) gives

| \sin \frac{\sqrt{3} z}{2} | > \frac{1}{4} e^{\frac{\sqrt{3}}{2} | ℑ z |}, | \sin \frac{\sqrt{3} ω z}{2} | > \frac{1}{4} e^{\frac{\sqrt{3}}{2} | ℑ ω z |}, | \sin \frac{\sqrt{3} ω^{2} z}{2} | > \frac{1}{4} e^{\frac{\sqrt{3}}{2} | ℑ ω^{2} z |},

λ \in C ∖ \cup_{n ⩾ 0} D_{n}

. The identity (3.5) gives

| Δ_{0} (λ) | > \frac{e^{\frac{\sqrt{3}}{2} (| ℑ z | + | ℑ ω z | + | ℑ ω^{2} z |)}}{24 \sqrt{3} | λ |} = \frac{e^{\frac{\sqrt{3}}{2} (\sqrt{3} ℜ z + ℑ z)}}{24 \sqrt{3} | λ |} = \frac{| e^{z (ω^{2} + 2)} |}{24 \sqrt{3} | λ |}, λ \in C_{+} ∖ \cup_{n \in Z} D_{n},

which yields (3.22).

(iii) Let $(λ, u) \in Λ_{+} \times B (ε)$ . Substituting the estimates (3.11) into the definition (3.2) we obtain

ϕ_{1} = (\begin{matrix} 1 + α_{11} & 1 + α_{12} & 1 + α_{13} \\ e^{ω z} (1 + α_{21}) & e^{ω^{2} z} (1 + α_{22}) & e^{z} (1 + α_{23}) \\ e^{2 ω z} (1 + α_{31}) & e^{2 ω^{2} z} (1 + α_{32}) & e^{2 z} (1 + α_{33}) \end{matrix}), | α_{j k} (λ) | ⩽ \frac{C ‖ u ‖}{| z |}, j, k = 1, 2, 3,

(3.24)

for some

C > 0

. The identity (3.3) yields

Δ = \frac{i det ϕ_{j}}{3 \sqrt{3} λ} = \frac{i e^{(ω^{2} + 2) z}}{3 \sqrt{3} λ} det (\begin{matrix} 1 + α_{11} & e^{- ω^{2} z} (1 + α_{12}) & e^{- 2 z} (1 + α_{13}) \\ e^{ω z} (1 + α_{21}) & 1 + α_{22} & e^{- z} (1 + α_{23}) \\ e^{2 ω z} (1 + α_{31}) & e^{ω^{2} z} (1 + α_{32}) & 1 + α_{33} \end{matrix}) .

The estimates (2.42) yield

max {ℜ (ω - ω^{2}) z, ℜ (ω^{2} - 1) z, ℜ (2 ω - 2) z, ℜ (ω + ω^{2} - 2) z, ℜ (2 ω - ω^{2} - 1) z} ⩽ 0,

then the estimates (3.24) give

Δ = \frac{i e^{(ω^{2} + 2) z}}{3 \sqrt{3} λ} (det (\begin{matrix} 1 & e^{- ω^{2} z} & e^{- 2 z} \\ e^{ω z} & 1 & e^{- z} \\ e^{2 ω z} & e^{ω^{2} z} & 1 \end{matrix}) + A) = Δ_{0} + \frac{i e^{(ω^{2} + 2) z} A}{3 \sqrt{3} λ}, | A (λ) | ⩽ \frac{C ‖ u ‖}{| z |},

(3.25)

for some

C > 0

. This identity yields

\frac{Δ}{Δ_{0}} - 1 = \frac{i e^{(ω^{2} + 2) z} A}{3 \sqrt{3} λ Δ_{0}} .

Then the estimates (3.22) and (3.25) give (3.23).

Now we prove counting results for the three-point eigenvalues.

Lemma 3.4

Let $u \in B (ε)$ . Then there is exactly one simple real eigenvalue $μ_{n}$ of the operator $L$ , given by (1.1), and exactly one simple real eigenvalue ${\tilde{μ}}_{n}$ of the operator $\tilde{L}$ , given by (1.17), in each domain $D_{n}, n \in Z_{0} = Z ∖ {0}$ . There are no other eigenvalues. Moreover,

{\tilde{μ}}_{n} (u) = - μ_{- n} (u_{*}) = μ_{n} (u^{-}), \forall n \in Z_{0},

(3.26)

where

u_{*} = (p, - q), u^{-} (x) = u (1 - x), u_{*}^{-} (x) = u_{*} (1 - x), x \in R .

(3.27)

Proof.

Let $λ$ belong to the contour $\partial D_{n}$ for some $n \in Z$ and let $u \in B (ε)$ . The estimate (3.23) and the symmetry $Δ (\bar{λ}) = \bar{Δ (λ)}$ imply

| \frac{Δ (λ)}{Δ_{0} (λ)} - 1 | < 1.

This yields

| Δ (λ) - Δ_{0} (λ) | = | Δ_{0} (λ) | | \frac{Δ (λ)}{Δ_{0} (λ)} - 1 | < | Δ_{0} (λ) |

on all contours. Hence, by Rouché’s theorem,

Δ

has as many zeros, as

Δ_{0}

in each of the domain

D_{n}, n \in Z

. Since

Δ_{0}

has exactly one simple zero in each

D_{n}, n \in Z ∖ {0}

, the function

Δ

has exactly one simple zero

μ_{n}

in each domain

D_{n}, n \in Z ∖ {0}

and there are no zeros in

D_{0}

. The zeros are simple and the function

Δ

is real on

R

, then all zeros are real. Combining this result with the result of Badanin and Korotyaev (2021), Lm 5.2 ii, we obtain that there are no other eigenvalues.

The transpose operator, given by (1.17), is equal to the operator $- L (u_{*})$ , then the spectra satisfy $σ (\tilde{L} (u)) = - σ (L (u_{*}))$ . Using ${\tilde{μ}}_{n} (u) \in D_{n}$ and $- μ_{- n} (u_{*}) \in D_{n}$ for all $n \in Z_{0}$ , we obtain the first identity in (3.26). Moreover, the operator $\tilde{L} (u^{-})$ is unitarily equivalent to the operator $L (u)$ , then $σ (\tilde{L} (u^{-})) = σ (L (u))$ . Using ${\tilde{μ}}_{n} (u^{-}) \in D_{n}$ and $μ_{n} (u) \in D_{n}$ for all $n \in Z_{0}$ , we obtain ${\tilde{μ}}_{n} (u^{-}) = μ_{n} (u)$ , which yields the second identity in (3.26).

3.4. Improved Asymptotics of the Function

Δ

Introduce the matrix-valued function $w (λ), λ \in Λ_{+}$ , by

w (λ) = (w_{j k} (λ))_{j, k = 1}^{3} = (\begin{matrix} w_{1} (0, λ) & w_{2} (0, λ) & w_{3} (0, λ) \\ w_{1} (1, λ) & w_{2} (1, λ) & w_{3} (1, λ) \\ w_{1} (2, λ) & w_{2} (2, λ) & w_{3} (2, λ) \end{matrix}),

(3.28)

where

w_{j}

are given by (3.12). Define the function

W = det (\begin{matrix} w_{11} & 1 & 1 \\ w_{21} e^{ω z} & e^{ω^{2} z} & e^{z} \\ w_{31} e^{2 ω z} & e^{2 ω^{2} z} & e^{2 z} \end{matrix}) + det (\begin{matrix} 1 & w_{12} & 1 \\ e^{ω z} & w_{22} e^{ω^{2} z} & e^{z} \\ e^{2 ω z} & w_{32} e^{2 ω^{2} z} & e^{2 z} \end{matrix}) + det (\begin{matrix} 1 & 1 & w_{13} \\ e^{ω z} & e^{ω^{2} z} & w_{23} e^{z} \\ e^{2 ω z} & e^{2 ω^{2} z} & w_{33} e^{2 z} \end{matrix}) .

(3.29)

The functions

w

and

W

are analytic on

Λ_{+}

Lemma 3.5

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then the function $Δ$ satisfies

Δ = Δ_{0} + \frac{i W}{9 \sqrt{3} z^{5}},

(3.30)

where

Δ_{0}

is given by (3.4).

Proof.

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then $\int_{0}^{1} Θ (s, λ) d s = Ω$ and the identities (3.12) yield

ϕ_{2} = ϕ_{0} + \frac{1}{3 z^{2}} (p (0) ϕ_{0} Ω + (\begin{matrix} w_{11} & w_{12} & w_{13} \\ w_{21} e^{ω z} & w_{22} e^{ω^{2} z} & w_{23} e^{z} \\ w_{31} e^{2 ω z} & w_{32} e^{2 ω^{2} z} & w_{33} e^{2 z} \end{matrix})),

(3.31)

where

Ω

is given by (2.16),

ϕ_{0}

has the form (3.4). The identities (3.31) and direct calculations give

det ϕ_{2} = det ϕ_{0} + \frac{1}{3 z^{2}} (p (0) det ϕ_{0} Tr Ω + W) .

The identities

Tr Ω = 0

and (3.3) imply (3.30).

Now we improve the asymptotics (3.23) of the function $Δ$ . The asymptotics will be expressed in terms of the function $β$ that we will now define. Introduce the matrix-valued function $ζ (λ), λ \in C$ , by

ζ (λ) = (ζ_{j k} (λ))_{j, k = 1}^{3} = (\begin{matrix} ζ_{1} (0, λ) & ζ_{2} (0, λ) & 0 \\ ζ_{1} (1, λ) & ζ_{2} (1, λ) & ζ_{3} (1, λ) \\ 0 & ζ_{2} (2, λ) & ζ_{3} (2, λ) \end{matrix}),

where

ζ_{j}

are given by (3.7) (note that, due to (3.10),

ζ_{3} (0, λ) = ζ_{1} (2, λ) = 0

). Define the function

β = det (\begin{matrix} ζ_{11} & 1 & 1 \\ ζ_{21} e^{ω z} & e^{ω^{2} z} & e^{z} \\ 0 & e^{2 ω^{2} z} & e^{2 z} \end{matrix}) + det (\begin{matrix} 1 & ζ_{12} & 1 \\ e^{ω z} & ζ_{22} e^{ω^{2} z} & e^{z} \\ e^{2 ω z} & ζ_{32} e^{2 ω^{2} z} & e^{2 z} \end{matrix}) + det (\begin{matrix} 1 & 1 & 0 \\ e^{ω z} & e^{ω^{2} z} & ζ_{23} e^{z} \\ e^{2 ω z} & e^{2 ω^{2} z} & ζ_{33} e^{2 z} \end{matrix}) .

(3.32)

The functions

ζ

and

β

are analytic on

C ∖ (- \infty, 0]

Introduce the functions $θ_{j} (λ), j = 1, \dots, 6$ , by

θ_{j} = \int_{0}^{1} e^{- i \sqrt{3} ω^{j} z s} v (s) d s, j = 1, 2, 3, θ_{4} (λ) = \bar{θ_{1} (\bar{λ})}, θ_{5} (λ) = \bar{θ_{2} (\bar{λ})}, θ_{6} (λ) = \bar{θ_{3} (\bar{λ})} .

(3.33)

The functions

θ_{j} (λ), j = 1, \dots, 6

, are analytic on

C ∖ (- \infty, 0]

. The definitions (3.10) and the identities

θ_{4} = {\bar{θ}}_{1}, θ_{5} = {\bar{θ}}_{2}, θ_{6} = {\bar{θ}}_{3}

yield on

R_{+}

ζ = (\begin{matrix} (1 + e^{- i \sqrt{3} ω^{2} z}) θ_{2} + ω^{2} (1 + e^{i \sqrt{3} z}) {\bar{θ}}_{3} & (1 + e^{i \sqrt{3} ω z}) {\bar{θ}}_{2} & 0 \\ θ_{2} + ω^{2} {\bar{θ}}_{3} & {\bar{θ}}_{2} - ω e^{i \sqrt{3} z} θ_{3} & - 2 ℜ (ω^{2} e^{i \sqrt{3} ω z} θ_{1}) \\ 0 & - ω e^{i \sqrt{3} z} (1 + e^{i \sqrt{3} z}) θ_{3} & - 2 ℜ (ω^{2} e^{i \sqrt{3} ω z} (1 + e^{i \sqrt{3} ω z}) θ_{1}) \end{matrix}) .

(3.34)

The identities

i \sqrt{3} ω z_{n}^{o} = - (\sqrt{3} + i) π n

and

i \sqrt{3} ω^{2} z_{n}^{o} = (\sqrt{3} - i) π n

give

θ_{1} (μ_{n}^{o}) = v_{n}^{+}, θ_{2} (μ_{n}^{o}) = v_{n}^{-}, θ_{3} (μ_{n}^{o}) = {\hat{v}}_{n},

(3.35)

where

{\hat{v}}_{n} = \int_{0}^{1} e^{- i 2 π n s} v (s) d s, v_{n}^{\pm} = \int_{0}^{1} e^{(\pm \sqrt{3} + i) π n s} v (s) d s .

(3.36)

We prove the following results.

Lemma 3.6

(i)

The function $β$ , given by (3.32), satisfies

\begin{aligned} β & = 2 i ℑ ((1 - e^{- i \sqrt{3} ω z} + e^{i \sqrt{3} ω^{2} z} - e^{- i \sqrt{3} z}) θ_{2} + ω (1 - e^{- i \sqrt{3} ω z} - e^{- i \sqrt{3} ω^{2} z} + e^{i \sqrt{3} z}) θ_{3} \\ - 2 e^{i \sqrt{3} z} ℜ (ω^{2} e^{i \sqrt{3} ω z} θ_{1}) - 2 e^{i \sqrt{3} ω^{2} z} ℜ (ω^{2} e^{i \sqrt{3} ω z} (1 + e^{i \sqrt{3} ω z}) θ_{1})), \end{aligned}

(3.37)

R_{+}

, where

θ_{j}

are given by (3.33). Moreover,

β (μ_{n}^{o}) = i 2 (- 1)^{n + 1} e^{\sqrt{3} π n} (ξ_{n}^{-})^{2} ℑ (ω {\hat{v}}_{n}), \forall n \in N,

(3.38)

where

ξ_{n}^{-}

is given by (2.11).

(ii)

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then the function $Δ$ satisfies

| Δ (λ) - Δ_{0} (λ) - \frac{i β (λ)}{9 \sqrt{3} z^{5}} | ⩽ \frac{C | e^{z (ω^{2} + 2)} | ‖ u ‖_{1}^{2}}{| z |^{6}},

(3.39)

for some

C > 0

, where

Δ_{0}

is given by (3.4).

Proof.

(i) The definition (3.32) gives

\begin{aligned} β & = e^{(ω^{2} + 2) z} (ζ_{11} + ζ_{22} + ζ_{33}) + e^{(ω + 2 ω^{2}) z} (ζ_{21} + ζ_{32}) + e^{(1 + 2 ω) z} (ζ_{12} + ζ_{23}) - e^{(ω^{2} + 2 ω) z} ζ_{22} \\ - e^{(ω + 2) z} (ζ_{21} + ζ_{12} + ζ_{33}) - e^{(1 + 2 ω^{2}) z} (ζ_{11} + ζ_{32} + ζ_{23}) . \end{aligned}

The identities (3.6) yield

β = A_{1} + A_{2} + A_{3} + (e^{i \sqrt{3} ω^{2} z} - e^{- i \sqrt{3} ω z}) ζ_{33} + (e^{i \sqrt{3} z} - e^{- i \sqrt{3} z}) ζ_{23},

(3.40)

where

\begin{aligned} A_{1} & = e^{i \sqrt{3} ω^{2} z} (ζ_{11} + ζ_{22}) - e^{- i \sqrt{3} ω z} (ζ_{21} + ζ_{12}), A_{2} = - e^{- i \sqrt{3} ω^{2} z} ζ_{22} + e^{i \sqrt{3} ω z} (ζ_{21} + ζ_{32}), \\ A_{3} & = e^{i \sqrt{3} z} ζ_{12} - e^{- i \sqrt{3} z} (ζ_{11} + ζ_{32}) . \end{aligned}

The identity (3.34) gives

\begin{aligned} A_{1} & = 2 i ℑ (e^{i \sqrt{3} ω^{2} z} θ_{2}) + 2 i ℑ θ_{2} - 2 i ℑ (e^{- i \sqrt{3} ω z} θ_{2}) - 2 i ℑ (ω e^{- i \sqrt{3} ω z} θ_{3}), \end{aligned}

\begin{aligned} A_{2} & = 2 i ℑ (e^{i \sqrt{3} ω z} θ_{2}) - 2 i ℑ (ω e^{- i \sqrt{3} ω^{2} z} θ_{3}), \end{aligned}

\begin{aligned} A_{3} & = - 2 i ℑ (e^{- i \sqrt{3} z} (1 + e^{- i \sqrt{3} ω^{2} z}) θ_{2}) + 2 i ℑ (ω (1 + e^{i \sqrt{3} z}) θ_{3}), \end{aligned}

for

λ > 0

. The identity (3.40) yields (3.37).

Let $λ = μ_{n}^{o} = z^{3}, z = z_{n}^{o}$ . Substituting the identities $i \sqrt{3} ω^{2} z = (\sqrt{3} - i) π n$ , $i \sqrt{3} ω z = - (\sqrt{3} + i) π n$ , and $e^{i \sqrt{3} z} = e^{i 2 π n} = 1$ into the identity (3.37) we obtain

β (μ_{n}^{o}) = 2 i (2 - (- 1)^{n} e^{\sqrt{3} π n} - (- 1)^{n} e^{- \sqrt{3} π n}) ℑ (ω θ_{3} (μ_{n}^{o})),

The definitions (2.11) imply

2 - (- 1)^{n} e^{\sqrt{3} π n} - (- 1)^{n} e^{- \sqrt{3} π n} = (- 1)^{n + 1} e^{\sqrt{3} π n} (ξ_{n}^{-})^{2},

which yields (3.38).

(ii) Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . The estimates (3.13) imply

| w_{j k} (λ) - ζ_{j k} (λ) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |},

(3.41)

for all

j, k = 1, 2, 3, λ \in Λ_{+}

, and for some

C > 0

. The definitions (3.32) and (3.29), give

| W (λ) - β (λ) | ⩽ \frac{C | e^{z (ω^{2} + 2)} | ‖ u ‖_{1}^{2}}{| z |},

for some

C > 0

. Then the identity (3.30) implies (3.39).

Introduce the function

γ (z) = e^{- \frac{3}{2} z} β (z^{3}),

and the domains

Π_{a} = {z \in C : ℜ z > a, | ℑ z | < 1}, Π_{a}^{+} = {z \in C_{+} : ℜ z > a, 0 < ℑ z < 1}, a \in R .

Lemma 3.7

Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then the function $γ (z)$ is analytic on the domain $Π_{1}$ and satisfies

| γ (z) | ⩽ C ‖ u ‖_{1}, z \in Π_{1},

(3.42)

γ (z_{n}^{o}) = i 2 (- 1)^{n + 1} (ξ_{n}^{-})^{2} ℑ (ω {\hat{v}}_{n}), n \in N,

(3.43)

| γ^{'} (z) | ⩽ C ‖ u ‖_{1} on [1, + \infty),

(3.44)

for some

C > 0

Proof.

The function $β$ is analytic on $C ∖ (- \infty, 0]$ , then $γ$ is analytic on the domain $Π_{1}$ . The identity (3.38) gives (3.43). Moreover, we have the estimates $ℜ (\pm i \sqrt{3} ω^{j} z) ⩽ \frac{3}{2} ℜ z$ in $Π_{1}^{+}$ for all $j = 1, 2, 3$ . Then the estimates (3.8) and the definition (3.40) imply (3.42) on $Π_{1}^{+}$ . Moreover, the function $i γ$ is real on $R_{+}$ , then the symmetry principle extends the estimate (3.42) onto $Π_{1}$ . Cauchy’s formula yields (3.44).

3.5. Rough Eigenvalue Asymptotics

Introduce the entire functions

ϰ_{0} (z) = - i 3 \sqrt{3} λ e^{- \frac{3}{2} z} Δ_{0} (z^{3}), ϰ (z) = - i 3 \sqrt{3} λ e^{- \frac{3}{2} z} Δ (z^{3}),

(3.45)

where

Δ_{0}

and

Δ

are given by (3.5) and (1.5) respectively. Recall that

μ_{n}^{o} = (z_{n}^{o})^{3}, n \in Z ∖ {0}

, are zeros of the function

Δ_{0} (λ)

, and

μ_{n} = z_{n}^{3}, n \in Z ∖ {0}

, are zeros of the function

Δ (λ)

, then

ϰ_{0} (z_{n}^{o}) = 0, ϰ (z_{n}) = 0, \forall n \in N .

(3.46)

Now we determine a rough asymptotics of the eigenvalues.

Lemma 3.8

(i)

The function $ϰ_{0}$ satisfies

\begin{matrix} ϰ_{0}^{'} (z_{n}^{o}) = (- 1)^{n + 1} i \sqrt{3} (ξ_{n}^{-})^{2}, \forall n \in N, \end{matrix}

(3.47)

\begin{matrix} | ϰ_{0}^{″} (z) | ⩽ C on [1, + \infty), \end{matrix}

(3.48)

for some

C > 0

, where

ξ_{n}^{-}

is given by (2.11).

(ii)

Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then the eigenvalues $μ_{n}$ satisfy

| μ_{n}^{\frac{1}{3}} - z_{n}^{o} | ⩽ \frac{C ‖ u ‖_{1}}{n^{2}},

(3.49)

for all

n \in N

and for some

C > 0

Proof.

(i) The identities (3.5) and (3.46) give

ϰ_{0}^{'} (z_{n}^{o}) = {- i 3 \sqrt{3} e^{- \sqrt{3} π n} \frac{d (z^{3} Δ_{0} (z^{3}))}{d z} |}_{z = z_{n}^{o}} = - 2 i \sqrt{3} e^{- \sqrt{3} π n} ((- 1)^{n} \cosh (\sqrt{3} π n) - 1),

which yields (3.47). The identity (3.5) implies

| ϰ_{0} (z) | ⩽ C

on the strip

Π_{0}

for some

C > 0

. Cauchy’s formula yields (3.48).

(ii) The estimate (3.39) gives

Δ = Δ_{0} + \frac{e^{\frac{3}{2} z} α}{λ}, | α (λ) | ⩽ \frac{C ‖ u ‖_{1}}{| z |^{2}} .

(3.50)

Let

z_{n} = μ_{n}^{\frac{1}{3}}, n \in N

. Due to Lemma 3.4,

z_{n} = z_{n}^{o} + δ_{n} \in R, - 1 < δ_{n} < 1

. The identity (3.50) implies

ϰ (z_{n}) = ϰ_{0} (z_{n}) + α (μ_{n}) = ϰ_{0} (z_{n}^{o} + δ_{n}) + α (μ_{n}) = δ_{n} ϰ_{0}^{'} (z_{n}^{o}) + \frac{δ_{n}^{2}}{2} ϰ_{0}^{″} (z) + α (μ_{n}),

where

z \in (z_{n}^{o}, z_{n})

and we used the identity

ϰ_{0} (z_{n}^{o}) = 0

. The identity

ϰ (z_{n}) = 0

, see (3.46), gives

δ_{n} = - \frac{ϰ_{0}^{'} (z_{n}^{o})}{2} (1 - {(1 - \frac{2 ϰ_{0}^{″} (z) α (μ_{n})}{{ϰ_{0}^{'}}^{2} (z_{n}^{o})})}^{\frac{1}{2}}) .

The estimates

| 1 - (1 - ε)^{\frac{1}{2}} | ⩽ | ε |

for all

ε \in R

small enough and the relations (3.39), (3.47), and (3.48) give

| δ_{n} | ⩽ \frac{C ‖ u ‖_{1}}{n^{2}},

for all

n \in N

and for some

C > 0

, which yields (3.49).

3.6. Improvement of the Eigenvalues Asymptotics

The definitions (1.12) imply

\begin{aligned} a_{- n} (u_{*}) & = - \frac{{\hat{p}}_{s n}^{'}}{\sqrt{3}} - {\hat{q}}_{c n} = - a_{n} (u), b_{- n} (u_{*}) = \frac{{\hat{p}}_{c n}^{'}}{\sqrt{3}} - {\hat{q}}_{s n} = b_{n} (u), \\ a_{- n} (u_{*}^{-}) & = - a_{n} (u^{-}) = - a_{n} (u), b_{- n} (u_{*}^{-}) = b_{n} (u^{-}) = - b_{n} (u) . \end{aligned}

(3.51)

Now we determine asymptotics of the 3-point eigenvalues for the case

u \in B_{1} (ε)

and

{\hat{p}}_{0} = {\hat{q}}_{0} = 0

. Recall that if

u \in B_{1} (ε)

, then

μ_{n}, n \in N

, are zeros of the function

ϕ_{2}

Lemma 3.9

Let $u \in B_{1} (ε)$ , let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ , and let $z_{n} = μ_{n}^{\frac{1}{3}}, {\tilde{z}}_{n} = {\tilde{μ}}_{n}^{\frac{1}{3}}, n \in N$ . Then

max {| z_{n} - z_{n}^{o} + \frac{1}{(2 π n)^{2}} (a_{n} - \frac{b_{n}}{\sqrt{3}}) |, | {\tilde{z}}_{n} - z_{n}^{o} + \frac{1}{(2 π n)^{2}} (a_{n} + \frac{b_{n}}{\sqrt{3}}) |} ⩽ \frac{C ‖ u ‖_{1}^{2}}{n^{3}},

(3.52)

for some

C > 0

, where

a_{n}, b_{n}

are given by (1.12).

Proof.

Let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ and let $λ = μ_{n}, n \in N, z_{n} = λ^{\frac{1}{3}}$ . Then, due to the estimate (3.49),

z_{n} = z_{n}^{o} + δ_{n}, δ_{n} \in R, | δ_{n} | ⩽ \frac{C ‖ u ‖_{1}}{n^{2}}, z_{n}^{o} = \frac{2 π n}{\sqrt{3}} .

(3.53)

The estimate (3.39) implies

ϰ (z_{n}) = ϰ_{0} (z_{n}) + \frac{γ (z_{n})}{3 z_{n}^{2}} + \frac{α_{1} (n)}{n^{3}} = ϰ_{0} (z_{n}) + \frac{γ (z_{n})}{(2 π n)^{2}} + \frac{α_{2} (n)}{n^{3}}, | α_{j} (n) | ⩽ C ‖ u ‖_{1}^{2}, j = 1, 2,

(3.54)

where we used (3.42). The relations (3.47) and (3.48) and the estimate (3.53) imply

ϰ_{0} (z_{n}) = ϰ_{0} (z_{n}^{o} + δ_{n}) = δ_{n} ϰ_{0}^{'} (z_{n}^{o}) + \frac{δ_{n}^{2}}{2} ϰ_{0}^{″} (z) = i (- 1)^{n + 1} \sqrt{3} (ξ_{n}^{-})^{2} δ_{n} + \frac{α_{3} (n)}{n^{3}},

(3.55)

where

| α_{3} (n) | ⩽ C ‖ u ‖_{1}^{2}

z \in (z_{n}^{o}, z_{n})

ξ_{n}^{-}

are given by (2.11), and we used the identity

ϰ_{0} (z_{n}^{o}) = 0

. The relations (3.43) and (3.44) imply

γ (z_{n}) = γ (z_{n}^{o} + δ_{n}) = γ (z_{n}^{o}) + δ_{n} γ^{'} (z) = 2 i (- 1)^{n + 1} (ξ_{n}^{-})^{2} ℑ (ω {\hat{v}}_{n}) + \frac{α_{4} (n)}{n^{2}},

(3.56)

where

| α_{4} (n) | ⩽ C ‖ u ‖_{1}^{2}

z \in (z_{n}^{o}, z_{n})

. Substituting the asymptotics (3.55) and (3.56) into (3.54) we obtain

ϰ (z_{n}) = - i (- 1)^{n} \sqrt{3} (ξ_{n}^{-})^{2} δ_{n} - \frac{2 i (- 1)^{n}}{(2 π n)^{2}} (ξ_{n}^{-})^{2} ℑ (ω {\hat{v}}_{n}) + \frac{α_{5} (n)}{n^{3}},

where

| α_{5} (n) | ⩽ C ‖ u ‖_{1}^{2}

. The identity

ϰ (z_{n}) = 0

, see (3.46), gives

δ_{n} = - \frac{ℑ (ω {\hat{v}}_{n})}{2 \sqrt{3} (π n)^{2}} + \frac{α_{6} (n)}{n^{3}},

where

| α_{6} (n) | ⩽ C ‖ u ‖_{1}^{2}

. The identity

ℑ (ω {\hat{v}}_{n}) = - \frac{1}{2} ℑ ((1 - i \sqrt{3}) (\frac{i ({\hat{p}}_{c n}^{'} - i {\hat{p}}_{s n}^{'})}{\sqrt{3}} + {\hat{q}}_{c n} - i {\hat{q}}_{s n})) = \frac{\sqrt{3}}{2} (a_{n} - \frac{b_{n}}{\sqrt{3}}),

gives

δ_{n} = - \frac{1}{(2 π n)^{2}} (a_{n} - \frac{b_{n}}{\sqrt{3}}) + \frac{α_{6} (n)}{n^{3}},

then (3.53) gives the first estimate in (3.52). The identities (3.26), (3.51), and the first estimate in (3.52) give

{\tilde{z}}_{n} = - z_{- n} (u_{*}) = - z_{- n}^{o} + \frac{1}{(2 π n)^{2}} (a_{- n} (u_{*}) - \frac{b_{- n} (u_{*})}{\sqrt{3}}) + \frac{α_{7} (n)}{n^{3}} = z_{n}^{o} - \frac{1}{(2 π n)^{2}} (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{α_{7} (n)}{n^{3}},

where

| α_{7} (n) | ⩽ C ‖ u ‖_{1}^{2}

. This yields the second estimate in (3.52).

Lemma 3.9 gives Theorem 1.1.

Proof Proof of Theorem 1.1.

Due to Lemma 3.4, there is exactly one simple real eigenvalue $μ_{n}$ in each domain $D_{n}, n \in Z_{0} = Z ∖ {0}$ . The first estimate in (3.52) yields (1.11) for $n > 0$ . If $n < 0$ , then the identities (3.26) and (3.51) imply

μ_{n} (u) = - {\tilde{μ}}_{- n} (u_{*}) = - μ_{- n}^{o} + a_{- n} (u_{*}) + \frac{b_{- n} (u_{*})}{\sqrt{3}} + \frac{α_{8} (n)}{n} = μ_{n}^{o} - a_{n} (u) + \frac{b_{n} (u)}{\sqrt{3}} + \frac{α_{8} (n)}{n},

where

| α_{8} (n) | ⩽ C ‖ u ‖_{1}^{2}

. This yields (1.11) for

n < 0

4. Norming Constants

4.1. Two Representations of the Monodromy Matrix

In this section we determine asymptotics of the norming constants defined by (1.18). We first express the norming constants in terms of the multipliers and the fundamental solutions. Then we determine the asymptotics of the multipliers and the fundamental solutions. This gives us the asymptotic expressions of the norming constants.

Introduce the monodromy matrix $M (λ), λ \in C$ , by

M (λ) = Φ (1, λ),

where the fundamental matrix

Φ

is given by (2.1). Now we obtain two representations of the monodromy matrix as a product of bounded in

Λ_{+}

matrices and a diagonal matrix, the first representation (4.1) for the case

p, q \in L^{2} (T)

and the second representation (4.3) for the case

p^{'}, q \in L^{2} (T)

. Assume

N = 1

in Lemmas 2.4 and 2.5.

Corollary 4.1
(i)
Let $u \in B (\frac{1}{12})$ . Then the monodromy matrix satisfies
$M (λ) = Ψ_{1} (1, λ) Ψ_{1}^{- 1} (0, λ) = U_{1} (λ) e^{z Ω} G_{1} (λ) U_{1}^{- 1} (λ),$
(4.1)
for all $λ \in Λ_{+}$ , where $Ψ_{1}$ is given by (2.51),
$U_{1} (λ) = U_{1} (λ) X_{1} (1, λ), G_{1} (λ) = X_{1}^{- 1} (0, λ) X_{1} (1, λ),$
(4.2)
$U_{1}$ is given by (2.13), $X_{1}$ is defined in Lemma 2.4, the matrix-valued functions $U_{1}$ and $G_{1}$ are analytic in the domain $Λ_{+}$ , each of the functions $U_{1} (λ, \cdot)$ and $G_{1} (λ, \cdot), λ \in Λ_{+}$ , is analytic on $B (\frac{1}{12})$ .
(ii)
Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then
$M (λ) = Ψ_{2} (1, λ) Ψ_{2}^{- 1} (0, λ) = U_{2} (λ) e^{z Ω} G_{2} (λ) U_{2}^{- 1} (λ),$
(4.3)
where $Ψ_{2}$ is given by (2.67),
$U_{2} (λ) = U_{1} (λ) U_{2} (0, λ) X_{2} (1, λ), G_{2} (λ) = X_{2}^{- 1} (0, λ) X_{2} (1, λ),$
(4.4)
$U_{2}$ is given by (2.20), $X_{2}$ is defined in Lemma 2.5 (i).

Proof.
The identity (2.53) gives (4.1). If ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ , then the definition (2.25) implies $\int_{0}^{1} Θ (s, λ) d s = Ω$ . The identities (2.69) and $U_{2} (1, λ) = U_{2} (0, λ)$ give (4.3).
4.2. Estimates of the Monodromy Matrix

Below we need the following estimates for the monodromy matrix.

Lemma 4.2
(i)
Let $(λ, u) \in Λ_{+} \times B (\frac{1}{12})$ . Then the matrix-valued function $G_{1}$ , given by (4.2), satisfies
$| G_{1} (λ) - 1 1_{3} | ⩽ \frac{24 ‖ u ‖}{| z |} .$
(4.5)
(i)
Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ . Then the matrix-valued function $G_{2}$ , given by (4.4), satisfies
$\begin{matrix} | G_{2} (λ) - 1 1_{3} | ⩽ \frac{C ‖ u ‖_{1}}{| z |^{2}}, \end{matrix}$
(4.6)

$\begin{matrix} | G_{2} (λ) - 1 1_{3} - \frac{L_{1} (λ)}{z^{2}} - \frac{L_{2} (λ)}{z^{4}} | ⩽ \frac{C ‖ u ‖_{1}^{3}}{| z |^{6}}, \end{matrix}$
(4.7)
for some $C > 0$ , where
$L_{1} (λ) = R_{1} (1, λ) - R_{1} (0, λ),$
(4.8)

$L_{2} (λ) = R_{2} (1, λ) - R_{2} (0, λ) + R_{1}^{2} (0, λ) - R_{1} (0, λ) R_{1} (1, λ),$
(4.9)
$R_{1}$ and $R_{2}$ are given by (2.63). The diagonal entries of the matrix $L_{1}$ satisfy
$L_{1, j j} = 0, j = 1, 2, 3.$
(4.10)

Proof.
(i) Let $(λ, u) \in Λ_{+} \times B (\frac{1}{12})$ . The definition (4.2) implies
$| G_{1} (λ) - 1 1_{3} | ⩽ | X_{1}^{- 1} (0, λ) - 1 1_{3} | + | X_{1} (1, λ) - 1 1_{3} | + | X_{1}^{- 1} (0, λ) - 1 1_{3} | | X_{1} (1, λ) - 1 1_{3} | .$
The estimates (2.48) and (2.54) give (4.5).

(ii) Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ . Using the definition (4.4) and the estimates (2.62) and repeating the previous arguments we obtain (4.6). Furthermore, the estimates (2.62) imply
$\begin{aligned} max {| X_{2} (1, λ) - 1 1_{3} - \frac{R_{1} (1, λ)}{z^{2}} - \frac{R_{2} (1, λ)}{z^{4}} |, \\ | X_{2}^{- 1} (0, λ) - 1 1_{3} + \frac{R_{1} (0, λ)}{z^{2}} - \frac{R_{1}^{2} (0, λ) - R_{2} (0, λ)}{z^{4}} |} ⩽ \frac{C ‖ u ‖_{1}^{3}}{z^{6}} . \end{aligned}$
These estimates and the definition (4.4) yield (4.7), where $L_{1}, L_{2}$ are given by (4.8) and (4.9). The definitions (2.57) and (2.58) and the identity (2.26) give
$R_{1, j j} (1, λ) = 0, R_{1, j j} (0, λ) = - \int_{0}^{1} F_{2, j j} (s, λ) d s = 0.$
(4.11)
The identity (4.8) yields (4.10).
4.3. Multipliers

Due to (4.1), the multipliers $τ_{1}, τ_{2}$ , and $τ_{3}$ are eigenvalues of the matrix $e^{z Ω} G_{1} = (e^{z ω^{j}} G_{1, j k})_{j, k = 1}^{3}$ . Introduce Gershgorin’s discs on the $τ$ -plane

\begin{aligned} Γ_{1} & = {τ \in C : | τ - e^{z ω} G_{1, 11} | ⩽ (| G_{1, 12} | + | G_{1, 13} |) | e^{z ω} |}, \\ Γ_{2} & = {τ \in C : | τ - e^{z ω^{2}} G_{1, 22} | ⩽ (| G_{1, 21} | + | G_{1, 23} |) | e^{z ω^{2}} |}, \\ Γ_{3} & = {τ \in C : | τ - e^{z} G_{1, 33} | ⩽ (| G_{1, 31} | + | G_{1, 32} |) | e^{z} |} . \end{aligned}

(4.12)

Lemma 4.3

(i)

Let $u \in B (ε)$ . Then

Γ_{1} \cap Γ_{3} = \emptyset, as λ \in Λ_{+}, Γ_{2} \cap Γ_{3} = \emptyset, as λ \in D,

(4.13)

where

D

is given by (3.18).

(ii)

Let $(λ, u) \in D \times B (ε)$ . Then there exists exactly one simple multiplier $τ_{3}$ in the disc $Γ_{3}$ Moreover,

| τ_{3} (λ) - e^{z} | ⩽ \frac{C ‖ u ‖}{| z |} | e^{z} |,

(4.14)

for some

C > 0

. The function

τ_{3}

is analytic on the domain

D

. Moreover,

τ_{3} (λ, \cdot), λ \in D

, is analytic on the ball

B (ε)

. Furthermore, the function

τ_{3}

is real for real

λ

and satisfies

τ_{3} (λ) > 0, \forall λ > 1.

(4.15)

Proof.

(i) Discs do not intersect if the distance between their centers is greater than the sum of their radii. Thus, the first relation in (4.13) holds iff

| e^{z ω} G_{1, 11} - e^{z} G_{1, 33} | > (| G_{1, 12} | + | G_{1, 13} |) | e^{z ω} | + (| G_{1, 31} | + | G_{1, 32} |) | e^{z} |,

which is equivalent to

r_{1} := | e^{(ω - 1) z} G_{1, 11} - G_{1, 33} | - (| G_{1, 12} | + | G_{1, 13} |) | e^{(ω - 1) z} | - | G_{1, 31} | - | G_{1, 32} | > 0.

Let

(λ, u) \in Λ_{+} \times B (\frac{1}{12})

. The estimate (4.5) gives

| G_{1, j j} | ⩾ 1 - \frac{24 ‖ u ‖}{| z |} ⩾ 1 - 24 ‖ u ‖, | G_{1, j k} | ⩽ \frac{24 ‖ u ‖}{| z |} ⩽ 24 ‖ u ‖, j = 1, 2, 3, j \neq k,

and using the estimate (3.20) we obtain

\begin{aligned} | e^{(ω - 1) z} G_{1, 11} - G_{1, 33} | & ⩾ | G_{1, 33} | - | e^{(ω - 1) z} G_{1, 11} | ⩾ 1 - 24 ‖ u ‖ - e^{- \frac{3}{2}} (1 - 24 ‖ u ‖) \\ = 1 - e^{- \frac{3}{2}} - 24 ‖ u ‖ (1 + e^{- \frac{3}{2}}) . \end{aligned}

Moreover,

(| G_{1, 12} | + | G_{1, 13} |) | e^{(ω - 1) z} | + | G_{1, 31} | + | G_{1, 32} | ⩽ 48 ‖ u ‖ (1 + e^{- \frac{3}{2}}),

which yields

r_{1} ⩾ 1 - e^{- \frac{3}{2}} - 72 ‖ u ‖ (1 + e^{- \frac{3}{2}}) .

‖ u ‖ ⩽ ε

, then

r_{1} > 0

, which yields the first relation in (4.13).

Furthermore, $Γ_{2} \cap Γ_{3} = \emptyset$ iff

| e^{z ω^{2}} G_{1, 22} - e^{z} G_{1, 33} | > (| G_{1, 21} | + | G_{1, 23} |) | e^{z ω^{2}} | + (| G_{1, 31} | + | G_{1, 32} |) | e^{z} |,

which is equivalent to

r_{2} := | e^{(ω^{2} - 1) z} G_{1, 22} - G_{1, 33} | - (| G_{1, 21} | + | G_{1, 23} |) | e^{(ω^{2} - 1) z} | - | G_{1, 31} | - | G_{1, 32} | > 0.

Let

(λ, u) \in Λ_{+} \times B (\frac{1}{12})

. The estimates (4.5) and

| e^{(ω^{2} - 1) z} | ⩽ 1

give

(| G_{1, 21} | + | G_{1, 23} |) | e^{(ω^{2} - 1) z} | + | G_{1, 31} | + | G_{1, 32} | ⩽ \frac{48 ‖ u ‖}{| z |} ⩽ 48 ‖ u ‖ .

Moreover,

\begin{aligned} | e^{(ω^{2} - 1) z} G_{1, 22} - G_{1, 33} | & = | e^{(ω^{2} - 1) z} - 1 + e^{(ω^{2} - 1) z} (G_{1, 22} - 1) - (G_{1, 33} - 1) | \\ ⩾ | e^{(ω^{2} - 1) z} - 1 | - | e^{(ω^{2} - 1) z} | | G_{1, 22} - 1 | - | G_{1, 33} - 1 | ⩾ | e^{(ω^{2} - 1) z} - 1 | - 24 ‖ u ‖ . \end{aligned}

Thus

r_{2} ⩾ | e^{(ω^{2} - 1) z} - 1 | - 72 ‖ u ‖ .

(4.16)

The estimates (4.16) and (3.21) give

r_{2} ⩾ \frac{1}{3} - 72 ‖ u ‖ .

‖ u ‖ < ε

, then

r_{2} > 0,

which yields the second relation in (4.13).

(ii) Let $(λ, u) \in D \times B (ε)$ . Then the disc $Γ_{3}$ is separated from $Γ_{1}$ and $Γ_{2}$ . Due to Gershgorin’s theorem, see Horn and Johnson (1985), Ch 6.1, there exists exactly one simple multiplier $τ_{3}$ in the disc $Γ_{3}$ . The estimates

| τ_{3} (λ) - e^{z} | ⩽ | e^{z} | (| G_{1, 31} | + | G_{1, 32} | + | G_{1, 33} - 1 |) ⩽ \frac{C ‖ u ‖}{| z |} | e^{z} |

give (4.14). The matrix-valued function

M

is entire, then the functions

τ_{3}

is analytic on the domain

D

. Due to Lemma 2.1, the function

M (λ, \cdot), λ \in C

, is analytic on

H

, then each of the function

τ_{3} (λ, \cdot), λ \in D

, is analytic on the ball

B (ε)

. Moreover, if

λ

is real, then

τ_{1} = {\bar{τ}}_{2}

τ_{1}

and

τ_{2}

are real. The identity

τ_{1} τ_{2} τ_{3} = 1

provides that

τ_{3}

is also real. The estimate (4.14) gives

τ_{3} (λ) ⩾ e^{z} - \frac{C ‖ u ‖}{z} e^{z}, z > 0,

which yields (4.15).

4.4. Asymptotics of the Multiplier

τ_{3}

We determine asymptotics of the multiplier $τ_{3}$ in the case $p^{'}, q \in L^{2} (T)$ .

Lemma 4.4
Let $(λ, u) \in D \times B_{1} (ε)$ and let ${\hat{p}}_{o} = {\hat{q}}_{o} = 0$ . Then the multiplier $τ_{3}$ satisfies
$| e^{- z} τ_{3} (λ) - 1 | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{4}} .$
(4.17)
Proof.
The estimates (4.14) show that the multiplier $τ_{3}$ has the form
$τ_{3} = e^{z} (1 + δ),$
(4.18)
where $| δ | ⩽ \frac{C ‖ u ‖}{| z |}$ . Let $(λ, u) \in Λ_{+} \times B_{1} (ε)$ and let ${\hat{p}}_{o} = {\hat{q}}_{o} = 0$ . The identity (4.3) yields that $τ_{3}$ is an eigenvalue of the matrix $e^{z Ω} G_{2}$ , where $G_{2}$ is given by (4.4). Due to (4.6), (4.7), and (4.10), the matrix $G_{2}$ has the form
$G_{2} = (\begin{matrix} 1 + \frac{α_{11}}{z^{4}} & \frac{α_{12}}{z^{2}} & \frac{α_{13}}{z^{2}} \\ \frac{α_{21}}{z^{2}} & 1 + \frac{α_{22}}{z^{4}} & \frac{α_{23}}{z^{2}} \\ \frac{α_{31}}{z^{2}} & \frac{α_{32}}{z^{2}} & 1 + \frac{α_{33}}{z^{4}} \end{matrix}),$
(4.19)
where
$| α_{j j} (λ) | ⩽ C ‖ u ‖_{1}^{2}, | α_{j k} (λ) | ⩽ C ‖ u ‖_{1}, j, k = 1, 2, 3.$
(4.20)
The identity (4.18) implies
$det (e^{z Ω} G_{2} - τ_{3} 1 1_{3}) = e^{3 z} det (e^{z (Ω - 1 1_{3})} G_{2} - (1 + δ) 1 1_{3}) = e^{3 z} det (G_{3} - δ_{1} 1 1_{3}),$
where
$δ_{1} = δ - \frac{α_{33}}{z^{4}},$
(4.21)

$\begin{aligned} G_{3} = e^{z (Ω - 1 1_{3})} G_{2} - (1 + \frac{α_{33}}{z^{4}}) 1 1_{3} = (\begin{matrix} e^{(ω - 1) z} (1 + \frac{α_{11}}{z^{4}}) - 1 - \frac{α_{33}}{z^{4}} & e^{(ω^{2} - 1) z} \frac{α_{12}}{z^{2}} & \frac{α_{13}}{z^{2}} \\ e^{(ω - 1) z} \frac{α_{21}}{z^{2}} & e^{(ω^{2} - 1) z} (1 + \frac{α_{22}}{z^{4}}) - 1 - \frac{α_{33}}{z^{4}} & \frac{α_{23}}{z^{2}} \\ e^{(ω - 1) z} \frac{α_{31}}{z^{2}} & e^{(ω^{2} - 1) z} \frac{α_{32}}{z^{2}} & 0 \end{matrix}) . \end{aligned}$
Then $δ_{1}$ is the eigenvalue of the matrix $G_{3}$ and, due to Gershgorin’s theorem and (4.20), it satisfies the estimate
$| δ_{1} | ⩽ \frac{max {| α_{13} |, | α_{23} |}}{| z |^{2}} ⩽ \frac{C ‖ u ‖_{1}}{| z |^{2}},$
for some $C > 0$ . Moreover,
$0 = det (G_{3} - δ_{1} 1 1_{3}) = - δ_{1} (e^{(ω - 1) z} - 1) (e^{(ω^{2} - 1) z} - 1) + \frac{α}{z^{4}},$
where $| α (λ) | ⩽ C ‖ u ‖_{1}^{2}$ , for some $C > 0$ . If $λ \in D$ , then the estimates (3.20) and (3.21) give $| δ_{1} | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{4}}$ , for some $C > 0$ . The identity (4.21) and the estimate (4.20) yield $| δ | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{4}}$ , then (4.18) implies (4.17).
4.5. Eigenfunctions

Introduce the functions

η (t, λ) = det (\begin{matrix} φ_{2} (t, λ) & φ_{3} (t, λ) \\ φ_{2} (1, λ) & φ_{3} (1, λ) \end{matrix}) .

(4.22)

Note that

η (0, λ) = η (1, λ) = 0, η (2, λ) = - Δ (λ),

(4.23)

where

Δ

is given by (1.5).

Lemma 4.5

(i)

Let $u \in H$ and let $μ \in C$ be an eigenvalue of the operator $L$ . Then the function $η (t, μ), t \in [0, 2]$ , given by (4.22), is the corresponding eigenfunction. Moreover, $η^{'} (0, μ) \neq 0$ and

\frac{η^{'} (1, μ)}{η^{'} (0, μ)} = - \frac{{\tilde{φ}}_{3} (1, μ)}{φ_{3} (1, μ)},

(4.24)

for all

t \in [0, 2]

(ii)

Let $u \in B_{1} (ε)$ . Then the norming constants, given by (1.18), satisfy

h_{s n} = 8 (π n)^{2} \log ((- 1)^{n + 1} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} τ_{3}^{- \frac{1}{2}} ({\tilde{μ}}_{n})), \forall n \in N .

(4.25)

Proof.

(i) Recall that $μ$ is a zero of the entire function $Δ$ . The definition (4.22) shows that $η$ satisfies Equation (1.3) and the identities (4.23) yield $η (0, μ) = η (1, μ) = η (2, μ) = 0$ . Therefore, $η$ is an eigenfunction and $η^{'} (0, μ) \neq 0$ . The identity (2.6) yields

{\tilde{φ}}_{3} = det (\begin{matrix} φ_{2} & φ_{3} \\ φ_{2}^{'} & φ_{3}^{'} \end{matrix}) .

Then the definition (4.22) and the identities

φ_{2}^{'} |_{x = 0} = 1

and

φ_{3}^{'} |_{x = 0} = 0

imply

\frac{η^{'} (1, μ)}{η^{'} (0, μ)} = \frac{det (\begin{matrix} φ_{2}^{'} (1, μ) & φ_{3}^{'} (1, μ) \\ φ_{2} (1, μ) & φ_{3} (1, μ) \end{matrix})}{det (\begin{matrix} φ_{2}^{'} (0, μ) & φ_{3}^{'} (0, μ) \\ φ_{2} (1, μ) & φ_{3} (1, μ) \end{matrix})} = - \frac{{\tilde{φ}}_{3} (1, μ)}{φ_{3} (1, μ)},

which yields (4.24).

(ii) Introduce the function

\tilde{η} (t, λ) = det (\begin{matrix} {\tilde{φ}}_{2} (t, λ) & {\tilde{φ}}_{3} (t, λ) \\ {\tilde{φ}}_{2} (1, λ) & {\tilde{φ}}_{3} (1, λ) \end{matrix}) .

Then

\tilde{η} (t, {\tilde{μ}}_{n})

is an eigenfunction of the operator

\tilde{L}

corresponding to the eigenvalue

{\tilde{μ}}_{n}

. The definition (1.18) yields

h_{s n} = 8 (π n)^{2} \log ((- 1)^{n} \frac{{\tilde{η}}^{'} (1, {\tilde{μ}}_{n})}{{\tilde{η}}^{'} (0, {\tilde{μ}}_{n})} τ_{3}^{- \frac{1}{2}} ({\tilde{μ}}_{n}))

for all

n \in N

. The identity (4.24) gives (4.25).

4.6. Sharp Asymptotics of the Fundamental Solution

In the following lemma, we determine the asymptotics of the fundamental solutions.

Lemma 4.6
Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then (i)
The fundamental solution $φ_{3}$ to Equation (1.3) satisfies
$| e^{- z} (φ_{3} (1, λ) - φ_{3}^{0} (1, λ) - \frac{2 p (0)}{3 λ} φ_{2}^{0} (1, λ) - \frac{b (λ)}{9 z^{4}}) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{6}},$
(4.26)
for all $λ \in Π_{1}$ and for some $C > 0$ , where $φ_{j}^{0}$ have the form (2.7),
$b = - e^{ω z} θ_{3} - e^{ω^{2} z} θ_{6} - ω e^{ω z} θ_{4} - ω e^{z} θ_{2} - ω^{2} e^{ω^{2} z} θ_{1} - ω^{2} e^{z} θ_{5},$
(4.27)
$θ_{j}$ are given by (3.33). Moreover, if $n \in N$ , then
$b (μ_{n}^{o}) = 2 e^{z_{n}^{o}} ℜ b_{n}, b_{n} = (- 1)^{n + 1} e^{- \sqrt{3} π n} {\hat{v}}_{n} - ω v_{n}^{-} - ω^{2} (- 1)^{n} e^{- \sqrt{3} π n} v_{n}^{+},$
(4.28)
where ${\hat{v}}_{n}, v_{n}^{-}, v_{n}^{+}$ are given by (3.36).
(ii)
The fundamental solution ${\tilde{φ}}_{3}$ to Equation (2.4) satisfies
$| e^{ω z} ({\tilde{φ}}_{3} (1, λ) - {\tilde{φ}}_{3}^{0} (1, λ) + \frac{2 p (0)}{3 λ} {\tilde{φ}}_{2}^{0} (1, λ) + \frac{\tilde{b} (λ)}{9 z^{4}}) | ⩽ \frac{C ‖ u ‖_{1}^{2}}{| z |^{6}},$
(4.29)
for all $λ \in Π_{1}$ and for some $C > 0$ , where ${\tilde{φ}}_{j}^{0}$ have the form (2.7),
$\tilde{b} = - e^{- ω^{2} z} θ_{3} - e^{- ω z} θ_{6} - ω e^{- z} θ_{4} - ω e^{- ω z} θ_{2} - ω^{2} e^{- z} θ_{1} - ω^{2} e^{- ω^{2} z} θ_{5} .$
(4.30)
Moreover, if $n \in N$ , then
$\tilde{b} (μ_{n}^{o}) = 2 (- 1)^{n + 1} e^{\frac{π n}{\sqrt{3}}} ℜ {\tilde{b}}_{n}, {\tilde{b}}_{n} = {\hat{v}}_{n} + ω v_{n}^{-} + ω^{2} (- 1)^{n} e^{- \sqrt{3} π n} v_{n}^{+} .$
(4.31)

Proof.
Let $u \in B_{1} (ε)$ , ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ , and let $λ \in Λ_{+}$ . The identities (2.67) and (4.3) give
$M (λ) = U_{1} (λ) S (1, λ) e^{z Ω} S^{- 1} (0, λ) U_{1}^{- 1} (λ), S (x, λ) = U_{2} (x, λ) X_{2} (x, λ) .$
(4.32)
This identity implies
$φ_{3} (1, λ) = M_{13} (λ) = \frac{1}{3 z^{2}} \sum_{j, k, l = 1}^{3} ω^{l} e^{z ω^{k}} S_{j k} (1, λ) (S^{- 1})_{k l} (0, λ),$
(4.33)
where we used the identities $U_{1, 1 j} (λ) = 1$ , $(U_{1}^{- 1})_{l 3} (λ) = \frac{ω^{l}}{3 z^{2}}$ , see (2.13). The definition (2.20) and the estimates (2.62) and (2.64) imply
$\begin{matrix} S (x, λ) = (1 1_{3} + \frac{p (x) Ω_{3}}{3 z^{2}}) (1 1_{3} + \frac{1}{3 z^{2}} \int_{0}^{1} N (x, s, λ) d s + \frac{α_{1} (x, λ)}{z^{4}}) = 1 1_{3} + \frac{W (x, λ)}{3 z^{2}} + \frac{α_{2} (x, λ)}{z^{4}}, \end{matrix}$
(4.34)

$\begin{matrix} S^{- 1} (x, λ) = 1 1_{3} - \frac{W (x, λ)}{3 z^{2}} + \frac{α_{3} (x, λ)}{z^{4}}, \end{matrix}$
(4.35)
where $N$ is given by (2.60), $sup_{x \in [0, 1]} | α_{j} (x, λ) | ⩽ C ‖ u ‖_{1}^{2}, j = 1, 2, 3$ , and
$W (x, λ) = p (x) Ω_{3} + \int_{0}^{1} N (x, s, λ) d s .$
(4.36)
These asymptotics yield
$S_{j k} (1, λ) (S^{- 1})_{k l} (0, λ) = δ_{j k} δ_{k l} + \frac{δ_{k l} W_{j k} (1, λ) - δ_{j k} W_{k l} (0, λ)}{3 z^{2}} + \frac{α_{6} (λ)}{z^{4}}$
where $| α_{6} (λ) | ⩽ C ‖ u ‖_{1}^{2}$ . Substituting these asymptotics into (4.33) and using (2.7) and the estimates $ℜ (ω^{k} - 1) ⩽ 0, k = 1, 2, 3$ , we obtain
$φ_{3} (1, λ) = φ_{3}^{0} (1, λ) + \frac{1}{9 z^{4}} (\sum_{j, l = 1}^{3} e^{z ω^{j}} W_{j l} (λ) + \frac{e^{z} α_{7} (λ)}{z^{2}}), | α_{7} (λ) | ⩽ C ‖ u ‖_{1}^{2},$
(4.37)
where
$\begin{matrix} W (λ) = Ω W^{⊤} (1, λ) - W (0, λ) Ω = p (0) A + B (λ), \end{matrix}$

$\begin{matrix} A = Ω Ω_{3}^{⊤} - Ω_{3} Ω = \frac{2 i}{\sqrt{3}} (\begin{matrix} 0 & - 1 & ω \\ 1 & 0 & - ω^{2} \\ - ω & ω^{2} & 0 \end{matrix}), \end{matrix}$
(4.38)

$\begin{matrix} B (λ) = \int_{0}^{1} (Ω N^{⊤} (1, s, λ) - N (0, s, λ) Ω) d s, \end{matrix}$
(4.39)
where $N$ is given by (2.60), here we used (4.36) and (2.21). The definition (4.39) gives
$B_{j l} (λ) = \int_{0}^{1} (ω^{j} N_{l j} (1, s, λ) - ω^{l} N_{j l} (0, s, λ)) d s, l, j = 1, 2, 3.$
(4.40)
The identity (3.9) implies
$N_{j l} (0, s, λ) = N_{l j} (1, s, λ) = 0, 0 ⩽ j ⩽ l ⩽ 3, s \in [0, 1] .$
(4.41)
Then $B_{j l} = 0, 0 ⩽ j ⩽ l ⩽ 3$ . The identities (3.9), (3.33), and (4.40) give
$B_{21} = - e^{i \sqrt{3} z} θ_{3} - θ_{6}, B_{31} = - ω e^{- i \sqrt{3} ω^{2} z} θ_{4} - ω θ_{2}, B_{32} = - ω^{2} e^{i \sqrt{3} ω z} θ_{1} - ω^{2} θ_{5} .$
(4.42)
Then the identity (4.37) implies
$φ_{3} (1, λ) = φ_{3}^{0} (1, λ) + \frac{1}{9 z^{4}} (p (0) a (λ) + b (λ) + \frac{e^{z} α_{7} (λ)}{z^{2}}),$
(4.43)
where
$a = \sum_{j, l = 1}^{3} e^{ω^{j} z} A_{j l}, b = \sum_{j, l = 1}^{3} e^{ω^{j} z} B_{j l} .$
(4.44)
The identities (4.38), (3.6), and (2.7) yield
$a (λ) = \frac{2 i}{\sqrt{3}} ((1 - ω^{2}) e^{z ω^{2}} + (ω - 1) e^{z ω} - (ω - ω^{2}) e^{z}) = 2 (ω^{2} e^{z ω} + ω e^{z ω^{2}} + e^{z}) = 6 z φ_{2}^{0} (1, λ) .$
Substituting this identity into (4.43) we obtain (4.26) for $λ \in Π_{1}^{+}$ . The identity (4.44) yields
$b = e^{ω^{2} z} B_{21} + e^{z} (B_{31} + B_{32}) .$
The identities (4.42) and (3.6) give (4.27).

If $λ > 0$ , then we have $θ_{2} = {\bar{θ}}_{5}, θ_{6} = {\bar{θ}}_{3}, θ_{1} = {\bar{θ}}_{4}$ . The identity (4.27) gives
$b = - 2 ℜ (e^{ω z} θ_{3} + ω^{2} e^{ω^{2} z} θ_{1} + ω e^{z} θ_{2}) .$
(4.45)
The function $b$ is analytic on $C ∖ (- \infty, 0]$ and real on $R_{+}$ , then the symmetry principle extends the estimate (4.26) onto $Π_{1}$ .

Substituting the identities (3.35) and the identities $e^{ω z_{n}^{o}} = e^{ω^{2} z_{n}^{o}} = (- 1)^{n} e^{- \frac{π n}{\sqrt{3}}}$ into (4.45) we obtain (4.28).

(ii) The identities (2.6) and (4.32) imply
$\tilde{M} (λ) = J (M^{- 1} (λ))^{⊤} J = J (U_{1}^{- 1} (λ))^{⊤} (S^{- 1} (1, λ))^{⊤} e^{- Ω z} S^{⊤} (0, λ) U_{1}^{⊤} (λ) J .$
This yields
${\tilde{φ}}_{3} (1, λ) = {\tilde{M}}_{13} (λ) = \frac{1}{3 z^{2}} \sum_{j, k, l = 1}^{3} ω^{j} e^{- ω^{k} z} (S^{- 1} (1, λ))_{k j} S_{l k} (0, λ),$
where we used the identities $J_{j k} = (- 1)^{j + 1} δ_{j, 4 - k}$ , $U_{1, 1 j} (λ) = 1$ , $(U_{1}^{- 1})_{l 3} (λ) = \frac{ω^{l}}{3 z^{2}}$ , see (2.6) and (2.13). The asymptotics (4.34) and (4.35) give
$\begin{aligned} {\tilde{φ}}_{3} (1, λ) & = {\tilde{φ}}_{3}^{0} (1, λ) + \frac{1}{9 z^{4}} \sum_{j, l = 1}^{3} e^{- ω^{j} z} (ω^{j} W_{l j} (0, λ) - ω^{l} W_{j l} (1, λ) + \frac{α_{8} (λ)}{z^{2}}) \\ = {\tilde{φ}}_{3}^{0} (1, λ) - \frac{1}{9 z^{4}} \sum_{j, l = 1}^{3} e^{- ω^{j} z} (W_{l j} (λ) + \frac{α_{8} (λ)}{z^{2}}) \end{aligned}$
where $| α_{8} (λ) | ⩽ C ‖ u ‖_{1}^{2}$ . The definition (4.36) and the estimates $ℜ (- z) ⩽ ℜ (- ω^{2} z) ⩽ ℜ (- ω z)$ imply
${\tilde{φ}}_{3} (1, λ) = {\tilde{φ}}_{3}^{0} (1, λ) - \frac{1}{9 z^{4}} (p (0) \tilde{a} (λ) + \tilde{b} (λ) + \frac{e^{- ω z} α_{9} (λ)}{z^{2}}),$
(4.46)
where $| α_{9} (λ) | ⩽ C ‖ u ‖_{1}^{2}$ ,
$\tilde{a} = \sum_{j = 1}^{3} e^{- ω^{j} z} \sum_{l = 1}^{3} A_{l j}, \tilde{b} = \sum_{j, l = 1}^{3} e^{- ω^{j} z} B_{l j} .$
The identities (2.7) and (4.38) yield
$\tilde{a} (λ) = \frac{2 i}{\sqrt{3}} ((1 - ω) e^{- ω z} + (ω^{2} - 1) e^{- ω^{2} z} + (ω - ω^{2}) e^{- z}) = - 2 (ω^{2} e^{- ω z} + ω e^{- ω^{2} z} + e^{- z}) = 6 z {\tilde{φ}}_{2}^{0} (1, λ),$
where we used the identities (3.6) and the definitions (2.7). Then the asymptotics (4.46) gives (4.29) for $λ \in Π_{1}^{+}$ . The identities (4.41) yield
$\tilde{b} = e^{- ω z} (B_{21} + B_{31}) + e^{- ω^{2} z} B_{32} .$
The identities (4.42) and (3.6) give (4.30).

If $λ > 0$ , then we have $θ_{2} = {\bar{θ}}_{5}, θ_{6} = {\bar{θ}}_{3}, θ_{1} = {\bar{θ}}_{4}$ . The identity (4.30) implies
$\tilde{b} = - 2 ℜ (e^{- ω^{2} z} θ_{3} + ω^{2} e^{- z} θ_{1} + ω e^{- ω z} θ_{2}) .$
The function $\tilde{b}$ is analytic on $C ∖ (- \infty, 0]$ and real on $R_{+}$ , then the symmetry principle extends the estimate (4.29) onto $Π_{1}$ . The identities (3.35) and $e^{- ω z_{n}^{o}} = e^{- ω^{2} z_{n}^{o}} = (- 1)^{n} e^{\frac{π n}{\sqrt{3}}}$ give (4.31).
4.7. Additional Asymptotics

In order to determine asymptotics of the norming constants we need the following results.

Lemma 4.7
Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then
$| (- 1)^{n + 1} e^{- \frac{π n}{\sqrt{3}}} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} - 1 + \frac{b_{n}}{2 \sqrt{3} (π n)^{2}} | ⩽ \frac{C ‖ u ‖_{1}^{2}}{n^{3}},$
(4.47)
for all $n \in N$ and for some $C > 0$ , where $b_{n}$ is given by (1.12).
Proof.
Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . The estimate (4.26) implies
$φ_{3} (1, λ) = φ_{3}^{0} (1, λ) + \frac{2 p (0)}{3 λ} φ_{2}^{0} (1, λ) + \frac{b (λ)}{9 z^{4}} + e^{z} \frac{α_{1} (λ)}{z^{6}}, | α_{1} (λ) | ⩽ C ‖ u ‖_{1}^{2},$
(4.48)
for all $λ \in Λ_{+}$ and for some $C > 0$ . Let $λ = {\tilde{μ}}_{n}$ for some $n \in N$ and let ${\tilde{z}}_{n} = λ^{\frac{1}{3}}$ . The estimate (3.49) gives
${\tilde{z}}_{n} = z_{n}^{o} + \frac{α_{2} (n)}{n^{2}}, z_{n}^{o} = \frac{2 π n}{\sqrt{3}}, | α_{2} (n) | ⩽ C ‖ u ‖_{1},$
(4.49)
for some $C > 0$ . The standard arguments yield
$φ_{3}^{0} (1, λ) = {φ_{3}^{0} (1, μ_{n}^{o}) + \frac{d φ_{3}^{0} (1, z^{3})}{d z} |}_{z = z_{n}^{o}} ({\tilde{z}}_{n} - z_{n}^{o}) + e^{z_{n}^{o}} \frac{α_{3} (n)}{n^{6}}, | α_{3} (n) | ⩽ C ‖ u ‖_{1}^{2} .$
The identities (2.9) imply
$φ_{3}^{0} (1, λ) = \frac{e^{z_{n}^{o}} ξ_{n}^{-}}{(2 π n)^{2}} + \frac{e^{z_{n}^{o}} ξ_{n}^{-}}{(2 π n)^{2}} (1 - \frac{\sqrt{3}}{π n}) ({\tilde{z}}_{n} - z_{n}^{o}) + e^{z_{n}^{o}} \frac{α_{3} (n)}{n^{6}},$
(4.50)
where $ξ_{n}^{-}$ is given by (2.11). Moreover, identity (2.8) implies
$φ_{2}^{0} (1, λ) = φ_{2}^{0} (1, μ_{n}^{o}) + e^{z_{n}^{o}} \frac{α_{4} (n)}{n^{3}} = \frac{ξ_{n}^{-} e^{z_{n}^{o}}}{2 \sqrt{3} π n} + e^{z_{n}^{o}} \frac{α_{4} (n)}{n^{3}}, | α_{4} (n) | ⩽ C ‖ u ‖_{1} .$
(4.51)
Furthermore, the definition (4.27) gives
$b (λ) = b (μ_{n}^{o}) + e^{z_{n}^{o}} \frac{α_{5} (n)}{n^{2}}, | α_{5} (n) | ⩽ C ‖ u ‖_{1}^{2} .$
(4.52)
Substituting the asymptotics (4.50)–(4.52) into (4.48), and using (4.28), we obtain
$φ_{3} (1, λ) = \frac{e^{z_{n}^{o}} ξ_{n}^{-}}{(2 π n)^{2}} + \frac{e^{z_{n}^{o}} ξ_{n}^{-}}{(2 π n)^{2}} (1 - \frac{\sqrt{3}}{π n}) ({\tilde{z}}_{n} - z_{n}^{o}) + \frac{e^{z_{n}^{o}} ξ_{n}^{-} p (0)}{8 (π n)^{4}} + \frac{2 e^{z_{n}^{o}} ℜ b_{n}}{(2 π n)^{4}} + e^{z_{n}^{o}} \frac{α_{6} (n)}{n^{6}},$
(4.53)
where $| α_{6} (n) | ⩽ C ‖ u ‖_{1}^{2}$ , for some $C > 0$ . The estimate (3.52) implies
${\tilde{z}}_{n} = z_{n}^{o} - \frac{1}{(2 π n)^{2}} (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{α_{7} (n)}{n^{3}},$
(4.54)
where $| α_{7} (n) | ⩽ C ‖ u ‖_{1}^{2}$ and $a_{n}, b_{n}$ are given by (1.12). Then
$φ_{3} (1, λ) = \frac{ξ_{n}^{-} e^{z_{n}^{o}}}{(2 π n)^{2}} (1 - \frac{1}{(2 π n)^{2}} (1 - \frac{\sqrt{3}}{π n}) (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{ℜ b_{n}}{2 (π n)^{2} ξ_{n}^{-}} + \frac{p (0)}{2 (π n)^{2}} + \frac{α_{8} (n)}{n^{3}}),$
(4.55)
where $| α_{8} (n) | ⩽ C ‖ u ‖_{1}^{2}$ , for some $C > 0$ .

The estimate (4.29) implies
${\tilde{φ}}_{3} (1, λ) = {\tilde{φ}}_{3}^{0} (1, λ) - \frac{2 p (0)}{3 λ} {\tilde{φ}}_{2}^{0} (1, λ) + \frac{\tilde{b} (λ)}{9 z^{4}} + \frac{e^{- ω z} {\tilde{α}}_{1} (λ)}{z^{6}},$
(4.56)
where $| {\tilde{α}}_{1} (λ) | ⩽ C ‖ u ‖_{1}^{2}$ , for all $λ \in Λ_{+}$ and for some $C > 0$ . Let $λ = {\tilde{μ}}_{n}$ for some $n \in N$ and let ${\tilde{z}}_{n} = λ^{\frac{1}{3}}$ . The estimate (4.49) yields
${\tilde{φ}}_{3}^{0} (1, λ) = {\tilde{φ}}_{3}^{0} (1, μ_{n}^{o}) + {\frac{d {\tilde{φ}}_{3}^{0} (1, z^{3})}{d z} |}_{z = z_{n}^{o}} ({\tilde{z}}_{n} - z_{n}^{o}) + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{2} (n)}{n^{6}}, | {\tilde{α}}_{2} (n) | ⩽ C ‖ u ‖_{1}^{2} .$
The identities (2.10) give
${\tilde{φ}}_{3}^{0} (1, λ) = \frac{(- 1)^{n + 1} e^{\frac{π n}{\sqrt{3}}} ξ_{n}^{-}}{(2 π n)^{2}} + \frac{(- 1)^{n} e^{\frac{π n}{\sqrt{3}}} ξ_{n}^{-}}{(2 π n)^{2}} (1 + \frac{\sqrt{3}}{π n}) ({\tilde{z}}_{n} - z_{n}^{o}) + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{2} (n)}{n^{6}} .$
(4.57)
Moreover, (2.8) implies
${\tilde{φ}}_{2}^{0} (1, λ) = {\tilde{φ}}_{2}^{0} (1, μ_{n}^{o}) + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{3} (n)}{n^{3}} = \frac{(- 1)^{n} e^{\frac{π n}{\sqrt{3}}} ξ_{n}^{-}}{2 \sqrt{3} π n} + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{3} (n)}{n^{3}}, | {\tilde{α}}_{3} (n) | ⩽ C ‖ u ‖_{1} .$
(4.58)
Furthermore, the definition (4.30) and the identity (4.31) give
$\tilde{b} (λ) = \tilde{b} (μ_{n}^{o}) + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{4} (n)}{n^{2}} = 2 (- 1)^{n} e^{\frac{π n}{\sqrt{3}}} ℜ {\tilde{b}}_{n} + e^{\frac{π n}{\sqrt{3}}} \frac{{\tilde{α}}_{4} (n)}{n^{2}}, | {\tilde{α}}_{4} (n) | ⩽ C ‖ u ‖_{1}^{2} .$
(4.59)
Substituting the asymptotics (4.57)–(4.59), and (4.54) into (4.56) we obtain
${\tilde{φ}}_{3} (1, λ) = \frac{(- 1)^{n + 1} ξ_{n}^{-} e^{\frac{π n}{\sqrt{3}}}}{(2 π n)^{2}} (1 + \frac{1}{(2 π n)^{2}} (1 + \frac{\sqrt{3}}{π n}) (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{p (0)}{2 (π n)^{2}} - \frac{2 ℜ {\tilde{b}}_{n}}{(2 π n)^{2} ξ_{n}^{-}} + \frac{{\tilde{α}}_{5} (n)}{n^{3}}),$
(4.60)
where $| {\tilde{α}}_{5} (n) | ⩽ C ‖ u ‖_{1}^{2}$ .

The asymptotics (4.55) and (4.60) give
$(- 1)^{n + 1} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} = e^{\frac{π n}{\sqrt{3}}} (1 - \frac{1}{2 (π n)^{2}} (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{ℜ (b_{n} + {\tilde{b}}_{n})}{2 (π n)^{2} ξ_{n}^{-}} + \frac{{\tilde{α}}_{6} (n)}{n^{3}}),$
(4.61)
where $| {\tilde{α}}_{6} (n) | ⩽ C ‖ u ‖_{1}^{2}$ . The definitions (4.28) and (4.31) imply
$ℜ (b_{n} + {\tilde{b}}_{n}) = ξ_{n}^{-} ℜ {\hat{v}}_{n} = ξ_{n}^{-} ℜ (\frac{i {\hat{p}}_{n}^{'}}{\sqrt{3}} + {\hat{q}}_{n}) = ξ_{n}^{-} a_{n} .$
Substituting this identity into (4.61) we obtain
$(- 1)^{n + 1} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} = e^{\frac{π n}{\sqrt{3}}} (1 - \frac{b_{n}}{2 \sqrt{3} (π n)^{2}} + \frac{{\tilde{α}}_{6} (n)}{n^{3}}),$
which yields (4.47).
4.8. Asymptotics of the Norming Constants

We are ready to prove Theorem 1.3.

Proof Proof of Theorem 1.3.

Let $λ = {\tilde{μ}}_{n}$ for some $n \in N$ and let ${\tilde{z}}_{n} = λ^{\frac{1}{3}}$ . The estimate (3.52) gives

{\tilde{z}}_{n} = z_{n}^{o} - \frac{1}{(2 π n)^{2}} (a_{n} + \frac{b_{n}}{\sqrt{3}}) + \frac{α_{1} (n)}{n^{3}},

where

| α_{1} (n) | ⩽ C ‖ u ‖_{1}^{2}

for some

C > 0

. Then the estimate (4.17) gives

τ_{3} ({\tilde{μ}}_{n}) = e^{{\tilde{z}}_{n}} (1 + \frac{α_{2} (n)}{n^{4}}) = e^{z_{n}^{o}} (1 - \frac{a_{n} + \frac{b_{n}}{\sqrt{3}}}{(2 π n)^{2}} + \frac{α_{3} (n)}{n^{3}}),

where

| α_{j} (n) | ⩽ C ‖ u ‖_{1}^{2}, j = 2, 3

. This yields

τ_{3}^{- \frac{1}{2}} ({\tilde{μ}}_{n}) = e^{- \frac{π n}{\sqrt{3}}} (1 + \frac{a_{n} + \frac{b_{n}}{\sqrt{3}}}{2 (2 π n)^{2}} + \frac{α_{4} (n)}{n^{3}}),

where

| α_{4} (n) | ⩽ C ‖ u ‖_{1}^{2}

. The estimate (4.47) gives

(- 1)^{n + 1} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} = e^{\frac{π n}{\sqrt{3}}} (1 - \frac{b_{n}}{2 \sqrt{3} (π n)^{2}} + \frac{α_{5} (n)}{n^{3}}), | α_{5} (n) | ⩽ C ‖ u ‖_{1}^{2} .

These asymptotics imply

(- 1)^{n + 1} \frac{φ_{3} (1, {\tilde{μ}}_{n})}{{\tilde{φ}}_{3} (1, {\tilde{μ}}_{n})} τ_{3}^{- \frac{1}{2}} ({\tilde{μ}}_{n}) = 1 - \frac{b_{n}}{2 \sqrt{3} (π n)^{2}} + \frac{a_{n} + \frac{b_{n}}{\sqrt{3}}}{2 (2 π n)^{2}} + \frac{α_{6} (n)}{n^{4}} = 1 + \frac{a_{n} - \sqrt{3} b_{n}}{2 (2 π n)^{2}} + \frac{α_{6} (n)}{n^{3}},

where

| α_{6} (n) | ⩽ C ‖ u ‖_{1}^{2}

. Substituting this asymptotics into (4.25) we obtain

h_{s n} = 8 (π n)^{2} \log (1 + \frac{a_{n} - \sqrt{3} b_{n}}{2 (2 π n)^{2}} + \frac{α_{6} (n)}{n^{3}}),

which yields (1.20).

In accordance with the identity (3.26), we introduce the norming constants $h_{s, - n}, n \in N$ by

h_{s, - n} (u) = - h_{s n} (u_{*}^{-}), n \in N .

(4.62)

Corollary 4.8

Let $u \in B_{1} (ε)$ and let ${\hat{p}}_{0} = {\hat{q}}_{0} = 0$ . Then the norming constants $h_{s n}$ satisfy the estimate (1.20) for all $n \in Z_{0}$ and for some $C > 0$ .

Proof.

Theorem 1.3 gives (1.20) for $n > 0$ . If $n < 0$ , then the identities (3.26) and (3.51) imply

h_{s n} (u) = - h_{s, - n} (u_{*}^{-}) = - a_{- n} (u_{*}^{-}) + \sqrt{3} b_{- n} (u_{*}^{-}) + \frac{α_{1} (n)}{n^{2}} = a_{n} (u) - \sqrt{3} b_{n} (u) + \frac{α (n)}{n},

where

| α (n) | ⩽ C ‖ u ‖_{1}^{2}

. This yields (1.20) for

n < 0

Footnotes

Acknowledgements

The authors were supported by the RSF grant number 23-21-00023.

ORCID iD

Andrey Badanin

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

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Asymptotics of the Divisor for the Good Boussinesq Equation

Abstract

Keywords

1. Introduction and Main Results

1.1. Introduction

2.1. The Fundamental Matrix

3.1. Two Representations of the Characteristic Function Δ

4. Norming Constants

4.1. Two Representations of the Monodromy Matrix

Proof Proof of Theorem 1.3.

Footnotes

Acknowledgements

ORCID iD

Funding

Declaration of Conflicting Interests

References

3.1. Two Representations of the Characteristic Function $Δ$