Semiclassical Birkhoff–Gustavson Normal Forms and Spectral Asymptotics for Nearly Resonant Schrödinger Operators

Abstract

The concept of near resonances for harmonic approximations of semiclassical Schrödinger operators is introduced and explored. Combined with a natural extension of the Birkhoff–Gustavson normal form, we obtain formulas for approaching the discrete spectrum of such operators which are both accurate and easy to implement. We apply the theory to the physically important case of the near Fermi (i.e. $1 : 2$ ) resonance, for which we propose explicit expressions and numerical computations.

Keywords

Schrödinger operators harmonic approximations near resonances Birkhoff–Gustavson normal form near Fermi resonance discrete spectrum

1. Introduction and Motivation

We are interested in the description of the discrete spectrum of a semiclassical Schrödinger operator ${\hat{P}}_{ε}$

{\hat{P}}_{ε} (ℏ) := - \frac{ℏ^{2}}{2} Δ + V_{ε} (x),

(1)

where

V_{ε}

is a smooth confining potential on

R^{n}

depending smoothly on a small parameter

ε \in R

, and

ℏ > 0

is a small parameter.

More precisely, by “confining potential” we mean that there exist some real value $E_{\infty} > 0$ and a small $ε_{0} > 0$ such that for all $E < E_{\infty}$ , the region $⋃_{| ε | \leq ε_{0}} V_{ε}^{- 1} ((- \infty, E))$ is bounded in $R^{n}$ . Moreover, we also assume that $V_{ε}$ grows at most polynomially at infinity, in the sense of the usual pseudo-differential symbol classes, uniformly with respect to $ε$ :

\begin{aligned} \exists m, \forall N \in N, \exists C_{N}, \forall | ε | \leq ε_{0}, \forall α \in N^{n} s.t. | α | \leq N, \\ \forall x \in R^{n} | \partial^{α} V_{ε} (x) | \leq C_{N} (1 + {‖ x ‖}^{2})^{m / 2} . \end{aligned}

This ensures that

{\hat{P}}_{ε}

has discrete spectrum in

(- \infty, E)

Let $E_{0, ε} = min V_{ε}$ . We will assume that this minimum is reached at a unique point and is non-degenerate; hence when studying the spectrum of $P_{ε}$ near $E_{0, ε}$ , it is natural to consider the harmonic approximation of $P_{ε}$ . As we will see in Section 2, up to an error of order $O (ε^{\infty})$ , we may assume that this approximation is smooth in $ε$ . Let us first consider $ε = 0$ . In the harmonic approximation of $P_{0}$ , which is the quantization of a quadratic Hamiltonian of the form

H_{2} (x, ξ) = \sum_{j = 1}^{n} \frac{ω_{j}}{2} (x_{j}^{2} + ξ_{j}^{2}),

two extreme cases may occur. Either the frequencies

(ω_{1}, \dots, ω_{n})

are independent over

Q

(this is the non-resonant case), or they are, up to some common multiple, all integers. Of course, intermediate cases may happen, see Definition 3.4. In the first case, a well-known result of Birkhoff (1927; following Poincaré) states that the full symbol

H

P_{0}

, which is a perturbation of

H_{2}

, is formally completely integrable: there are canonical coordinates

(x, ξ)

and a smooth map

f

such that

H (x, ξ) = f (I_{1}, \dots, I_{n}) + O (x, ξ)^{\infty},

(2)

where

I_{j}

is the action given by

I_{j} := \frac{1}{2} (x_{j}^{2} + ξ_{j}^{2}) .

Although the Birkhoff idea was soon used by physicists to deal with quantum Hamiltonians, a rigorous proof of the quantum validity of the Birkhoff normal form, in the semiclassical limit, is much more recent, see Sjöstrand (1992). This non-resonant case is stable under perturbations of order

O (ε)

if the quadratic term

H_{2}

is invariant, in the sense that one can write a combined Birkhoff normal form in all variables

(x, ξ, ε)

(and then a semiclassical Birkhoff normal form in

(x, ξ, ε, ℏ)

will hold). If one adds diophantine conditions to the frequencies

ω_{j}

, so that they become badly approximated by rationals, then it is expected that one can accommodate perturbations of the quadratic term, and even strengthen the result by using KAM stability, similarly to the case of diophantine tori in Hitrik et al. (2007). See also Lazutkin (1993).

In this paper, we focus on the resonant case, where the situation is quite different. For simplicity, we will always consider the fully resonant case, where all frequencies are integers, up to a common multiple. There is an extension of the Birkhoff normal form for the resonant case, which was worked out by Gustavson (1966) and Eckhardt (1986; although the general scheme was already known to Poincaré—see also Moser, 1958), where it was shown that, in addition to the completely integrable normal form (2), another formal series of resonant terms has to be considered, which makes the resulting series much more difficult to analyze (it will be, generically, non-integrable, see Duistermaat, 1984). The full semiclassical analysis of resonant harmonic approximations of general pseudodifferential operators was carried out in Charles and Vũ Ng $ọ$ c (2008).

When it comes to $ε$ perturbations, possibly affecting the harmonic term $H_{2}$ , one may argue that the resonant case is not generic: for most perturbations of a resonant Hamiltonian $H_{0}$ , the perturbed Hamiltonian $H_{ε}$ will be non-resonant. Thus, it is tempting to claim that, in most physical situations, one can restrict oneself to the set of non-resonant $ε$ and thus stick to the completely integrable normal form (2). However, this normal form is in general not convergent (Krikorian, 2022), and, as $ε \to 0$ , it is expected that the famous appearance of small denominators will make it more and more divergent, hindering the effectiveness of the approximation (unless the full Hamiltonian is known to be integrable and analytic, see Zung, 2005).

This phenomenon of near resonances has been recognized as crucial in molecular spectroscopy. Thus, for the study of the dynamics of highly excited vibrational states, Joyeux (1998) shows that the resonant Birkhoff–Gustavson procedure can yield more accurate results than the standard Birkhoff procedure, even when the Hamiltonian is not resonant. He considers the HCP molecule, called phosphatine, where the calculation of the fundamental frequencies leads to $ω_{1} \approx 1256$ (C–P stretch), $ω_{2} \approx 650$ (bend), and $ω_{3} \approx 3479 {c m}^{- 1}$ (C–H stretch). The Hamiltonian obtained by the non-resonant Birkhoff normal form (also called the Dunham expansion) is the formal series

H_{D} = \sum_{i} ω_{i} I_{i} + \sum_{i ⩽ j} x_{i j} I_{i} J_{j} + \sum_{i ⩽ j ⩽ k} y_{i j k} I_{i} J_{j} I_{k} + \dots

Joyeux computed levels of HCP up to

22, 000, {c m}^{- 1}

above the bottom of the well by truncating the series at various orders and compared the results to the exact quantum levels of HCP relative to the ground state. Using

ℓ^{1}

ℓ^{2}

, and

ℓ^{\infty}

norms to estimate the discrepancy with respect to the exact quantum computation, he observed a rapid divergence, see Joyeux (1998: Table 1), which limits the interest of this Hamiltonian to at most fourth or fifth order, which is not satisfactory. The article therefore concludes that the Dunham expansion is very far from being sufficient for whatever quantitative purpose.

Given that the relation between both fundamental frequencies $ω_{1}$ and $ω_{2}$ , namely $ω_{1} - 2 ω_{2} \approx - 44 {c m}^{- 1}$ is a near resonance relation, Joyeux proposed, as a next step, to take this resonance into account in the formal expansion, and to compare once again the results of the calculations of the energy levels of the HCP molecule. For this, he considered the following Hamiltonian:

\begin{aligned} H = H_{D} + \sum_{m ⩾ 1} H_{F}^{(m)}, \\ H_{F}^{(m)} = 2 I_{1}^{m / 2} I_{2}^{m} \cos (m φ_{1} - 2 m φ_{2}) \times (k^{(m)} + \sum_{i} k_{i}^{(m)} I_{i} + \sum_{i ⩽ j} k_{i j}^{(m)} I_{i} J_{j} + \dots), \end{aligned}

which he calls the Fermi resonance Hamiltonian (in physics or chemistry literature, the

2 : 1

resonance is traditionally called the Fermi resonance, see Section 3.2). The terms in

H_{F}

are precisely those given by the Birkhoff–Gustavson procedure in case of an exact

2 : 1

resonance. It is then noticed that the results of the computations of energy levels are much more accurate with this modified Hamiltonian.

From this study comes the motivation and main goal of our work, which is a description of the spectrum of semiclassical Schrödinger operators ${\hat{P}}_{ε} (ℏ)$ given in (1), for which the harmonic frequencies are close to resonance. For this purpose, we build on the paper by Charles and Vũ Ng $ọ$ c (2008), which gives precise semiclassical asymptotics for an exact, full resonance, by restricting the Hamiltonian to the eigenspaces of the resonant harmonic oscillator (whose dimensions tend to infinity in the semiclassical limit). Our main results are organized as follows:

In Section 2, we consider the first step of this work, which consists of transforming the initial Schrödinger Hamiltonian ${\hat{P}}_{ε}$ into a perturbation of the harmonic oscillator (Proposition 3.1). For this, we need a diagonalization result of symmetric real matrices depending smoothly on small parameter $ε$ (Theorem 2.1).

Section 3 has a double goal. On the one hand, we prove in Theorem 3.3 that the Birkhoff–Gustavson normal form (BGNF) theorem can be extended to handle Schrödinger operators ${\hat{P}}_{ε}$ depending on the parameter $ε$ (where the harmonic frequencies also depend on the parameter $ε$ ); on the other hand, we give an explicit construction of the BGNF in the near Fermi resonance, in Theorems 3.5 and 3.6.

In Section 4, we give the exact matrix representation of the “polyads” generated by the first non-trivial Birkhoff correction of the Fermi resonance, that is, the restriction of the quantum BGNF to the various eigenspaces of the resonant harmonic oscillator (Theorem 4.2).

Finally, in Section 5, we show how the theoretical study can lead to numerical schemes, and we propose, in the case of the Fermi resonance, a detailed numerical illustration of our results, by comparing the “exact” quantum spectrum of ${\hat{P}}_{ε}$ with the eigenvalues obtained via the $ε$ -Birkhoff–Gustavson procedure, at order 3.

2. Preparation: Smooth Diagonalization

In order to obtain the harmonic approximation of the Schrödinger operator (1), we need to diagonalize the Hessian of $V_{ε}$ at the critical point, in a smooth way. Because our aim is to deal with resonant eigenvalues, we cannot assume that eigenvalues are simple, and we will use the following general result, which is elementary but apparently not often found in the literature (we could not locate a reference).

Theorem 2.1
Let $A (ε)$ be a family of $n \times n$ real symmetric, respectively, Hermitian, matrices depending on a smooth (i.e. $C^{\infty}$ ) way on a small parameter $ε$ . Then there exists a smooth family of orthogonal, respectively, unitary, matrices $U (ε)$ and smooth functions $ε \mapsto λ_{j, ε} \in R$ , $j = 1, \dots, n$ such that
$U^{- 1} (ε) A (ε) U (ε) = d i a g (λ_{1, ε}, \dots, λ_{n, ε}) + O (ε^{\infty}) .$

Remark 2.2
This result means that, if we accept to replace the true eigenvalues with approximate eigenvalues which are close to the exact one to any order in $ε$ , then we may smoothly diagonalize the family $A (ε)$ . What may come as a surprise when one first encounters this kind of result is that, in general, it is not possible to smoothly diagonalize a smooth family of symmetric matrices in an exact way (unless all eigenvalues are simple). See for instance Example 5.3, section II-5 of the Kato book (Kato, 1980). On the other hand, positive results are available when the family is analytic, or if one only requires $C^{1}$ regularity for the eigenvalues, a result due to Rellich (1969), see also Kato (1980: Theorem 6.8 Section II-6).
Proof.
Let us treat the real symmetric case; the Hermitian case will be completely analogous. We reason by induction on the size of the matrix. The result obviously holds when $n = 1$ (without the $O (ε^{\infty})$ error term). We may assume that $A (0)$ is diagonal; let $μ_{1} < \dots < μ_{ℓ}$ be its eigenvalues, with multiplicities $d_{1}, \dots, d_{ℓ}$ . By the min–max formula, the eigenvalues of $A (ε)$ are continuous in $ε$ . Thus, there exists $ρ > 0$ such that the spectrum of $A (ε)$ is contained in $⋃_{j = 1}^{ℓ} B (μ_{j}, ρ)$ , and each ball $B_{j} := B (μ_{j}, ρ)$ contains $d_{j}$ eigenvalues (counted with multiplicity). This shows that the spectral projector on the generalized eigenspaces,
$P_{B_{j}} (ε) := - \frac{1}{2 π i} \int_{\partial B_{j}} (A (ε) - z)^{- 1} d z,$
is $C^{\infty}$ in $ε$ . It is now easy to find an orthonormal basis of the generalized eigenspace $E_{j} (ε) := P_{B_{j}} (R^{n})$ that depends smoothly on $ε$ . For instance, one can take a basis $B (0)$ of $E_{j} (0)$ ; for $ε$ small enough, the projection $P_{B_{j}} (ε) (B (0))$ is a basis of $E_{j} (ε)$ , which we may smoothly orthonormalize by the Gram–Schmidt algorithm. In this way we obtain a smooth bloc-diagonal decomposition: there exists a smooth unitary matrix $V (ε)$ such that
$V^{- 1} (ε) A (ε) V (ε) = (\begin{array}{ccc} A_{1} (ε) & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & A_{ℓ} (ε) \end{array}) .$
where $A_{j} (ε) = μ_{j} I + O (ε)$ is a real symmetric matrix. If $ℓ > 1$ we obtain the result by induction. Hence it only remains to consider the case of a unique generalized eigenspace of dimension $n$ : we have
$] A (ε) = μ_{1} I + O (ε) .$
Because the remainder $O (ε)$ is smooth, we can write it as $ε B_{1} (ε)$ , where $B_{1} (ε)$ is smooth (and real symmetric). Therefore the question is reduced to diagonalizing $B_{1} (ε)$ , and we may repeat the procedure. There are finally only two possibilities:
Either there exists $N > 0$ such that, after the $N$ th iteration, the remainder $B_{N} (0)$ possesses more than one generalized eigenspace. Then we may split them and obtain the result as above;

or for all $N > 0$ , $B_{N} (0)$ has only one generalized eigenspace.
In the second case, we have real constants $c_{0} = μ_{1}, c_{1}, c_{2}, \dots$ such that, for any $N \geq 0$ ,
$A (ε) = c_{0} I + c_{1} ε I + \dots + c_{N} ε^{N} I + O (ε^{N + 1}) .$
By the Borel lemma, there exists a $C^{\infty}$ function $λ (ε)$ whose Taylor series at $ε = 0$ is
$λ (ε) \sim c_{0} + c_{1} ε + c_{2} ε^{2} + \dots + c_{N} ε^{N} + \dots$
which means $\forall N, A (ε) - λ (ε) I = O (ε^{N})$ . Thus, $A (ε) = λ (ε) I + O (ε^{\infty})$ , which gives the result.
3. Birkhoff–Gustavson Normal Form in Near Resonance

In this section, we shall discuss the Birkhoff normal form procedure for Schrödinger operators ${\hat{P}}_{ε}$ which depend on small parameters $ℏ > 0$ and $ε$ , and we will then apply the general ideas to the near Fermi resonance.

3.1. $ε$ -Birkhoff–Gustavson Normal Form Theorem

On $L^{2} (R^{n})$ consider the Schrödinger operator

\hat{P_{ε}} = - \frac{ℏ^{2}}{2} Δ_{x} + V_{ε} (x),

where

Δ = Δ_{x}

is the

n

dimensional Laplacian and

V_{ε}

is a smooth real potential depending smoothly on

ε

. We wish to perform a local (and microlocal) analysis near the origin

x = 0

. To this effect, we assume that for

ε = 0

, the potential

V_{0}

has a non-degenerate minimum at the origin:

V_{0} (0) = 0, V_{0}^{'} (0) = 0, V_{0}^{″} (0) > 0.

Using the implicit function theorem to

F (ε, x) = V_{ε}^{'} (x)

, we obtain a smooth map

ε \mapsto x_{ε}

near the origin such that

x_{0} = 0

and

V_{ε}^{'} (x_{ε}) = 0

, for

ε > 0

small enough. So, there exists

ε_{0} > 0

such that for all

| ε | \leq ε_{0}

, the point

x_{ε}

is a non-degenerate minimum for

V_{ε}

. Using the translation

x \mapsto \tilde{x} = x - x_{ε}

, which yields a unitary map

τ_{ε}

L^{2} (R^{n})

, given by

τ_{ε} f (\tilde{x}) = f (\tilde{x} + x_{ε})

{\hat{P}}_{ε}

is transformed into

τ_{ε} {\hat{P}}_{ε} τ_{ε}^{- 1} = - \frac{ℏ^{2}}{2} Δ_{\tilde{x}} + W_{ε} (\tilde{x}),

where

W_{ε} (\tilde{x}) = V_{ε} (\tilde{x} + x_{ε})

. We have

W_{ε} (0) = V_{ε} (x_{ε}), W_{ε}^{'} (0) = V_{ε}^{'} (x_{ε}) = 0,

and the symmetric matrix

W_{ε}^{″} (0)

is positive definite, for

ε > 0

sufficiently small.

Using Theorem 2.1, we can smoothly diagonalize $W_{ε}^{″} (0)$ modulo $O (ε^{\infty})$ , via a change of variables $y = U_{ε}^{*} \tilde{x}$ , with $U_{ε} \in O (n)$ . Let $(ω_{1, ε}^{2}, \dots, ω_{n, ε}^{2})$ be its approximate eigenvalues: they are positive and depend in a $C^{\infty}$ way on $ε$ .

The change of variables $U_{ε}$ induces a unitary map $g_{ε}$ on $L^{2} (R^{n})$ given by $g_{ε} f (y) = f (U_{ε} y)$ . Since $U_{ε}$ is orthogonal, $g_{ε} Δ_{\tilde{x}} g_{ε}^{- 1} = Δ_{y}$ and hence the transformed operator is

g_{ε} τ_{ε} {\hat{P}}_{ε} τ_{ε}^{- 1} g_{ε}^{- 1} = - \frac{ℏ^{2}}{2} Δ_{y} + W_{ε} (U_{ε} y) .

Since

W_{ε} (\tilde{x}) = \frac{1}{2} ⟨ W_{ε}^{″} (0) \tilde{x}, \tilde{x} ⟩ + O ({\tilde{x}}^{3})

, we obtain

W_{ε} (U_{ε} y) = W_{ε} (0) + \frac{1}{2} \sum_{j = 1}^{n} ω_{j, ε}^{2} y_{j}^{2} + O (ε^{\infty} {‖ y ‖}^{2}) + O (y^{3}) .

(3)

The rescaling

y_{j} \mapsto {\tilde{y}}_{j} := \sqrt{ω_{j, ε}} y_{j}

, giving rise to the unitary map

Λ_{ε} : f \mapsto (\tilde{y} \mapsto \frac{1}{(ω_{1, ε} \dots ω_{n, ε})^{n / 2}} f (\frac{{\tilde{y}}_{1}}{\sqrt{ω_{1, ε}}}, \dots, \frac{{\tilde{y}}_{n}}{\sqrt{ω_{n, ε}}}))

transforms

\hat{P_{ε}}

into a perturbation of the harmonic oscillator

{\hat{H}}_{2, ε}

Λ_{ε} g_{ε} τ_{ε} {\hat{P}}_{ε} τ_{ε}^{- 1} g_{ε}^{- 1} Λ_{ε}^{- 1} = W_{ε} (0) + {\hat{H}}_{2, ε} + R_{ε} (\tilde{y}),

(4)

where

{\hat{H}}_{2, ε} = \sum_{j = 1}^{n} \frac{ω_{j, ε}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial {\tilde{y}}_{j}^{2}} + {\tilde{y}}_{j}^{2}) + O (ε^{\infty} {‖ \tilde{y} ‖}^{2}),

and

R_{ε} (\tilde{y})

is a smooth function of order

O ({\tilde{y}}^{3})

at the origin, uniformly with respect to

ε

From now on, to simplify notation, we switch back to $x$ , and assume that

{\hat{P}}_{ε} = W_{ε} (0) + {\hat{H}}_{2, ε} + R_{ε} (x),

with

{\hat{H}}_{2, ε} = \sum_{j = 1}^{n} \frac{ω_{j, ε}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{j}^{2}} + x_{j}^{2}) + O (ε^{\infty} ‖ x^{2} ‖) .

Let

{\hat{H}}_{2, 0} = \sum_{j = 1}^{n} \frac{ω_{j, 0}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{j}^{2}} + x_{j}^{2}),

The eigenvalues

ω_{j, ε}

being smooth in

ε

, we have

ω_{j, ε} = ω_{j, 0} + ε ω_{j, 1} + ε^{2} {\tilde{ω}}_{j, 2} (ε) .

Therefore,

{\hat{P}}_{ε} = W_{ε} (0) + {\hat{H}}_{2, 0} + ε {\hat{L}}_{2} + ε^{2} {\hat{M}}_{2, ε} + R_{ε} (x) + O (ε^{\infty} ‖ x^{2} ‖),

where

{\hat{L}}_{2} = \sum_{j = 1}^{n} \frac{ω_{j, 1}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{j}^{2}} + x_{j}^{2}) and {\hat{M}}_{2, ε} = \sum_{j = 1}^{n} \frac{{\tilde{ω}}_{j, 2} (ε)}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{j}^{2}} + x_{j}^{2}),

(5)

Because the symbol of

{\hat{L}}_{2}

is quadratic, it cannot be reduced by the usual Birkhoff–Gustavson procedure; in order to deal with this issue, we have to add

ε

to the set of formal variables, so that

ε {\hat{L}}_{2}

becomes of order 3.

Thus, we introduce the space

E = C [[x, ξ, ε, ℏ]] = C [[x_{1}, \dots, x_{n}, ξ_{1}, \dots, ξ_{n}, ε, ℏ]],

of formal power series of

(2 n + 2)

variables with complex coefficients, where the degree of the monomial

x^{α} ξ^{β} ε^{m} ℏ^{ℓ}

is defined to be

| α | + | β | + m + 2 ℓ

, and

α, β \in N^{n}, ℓ, m \in N

Let $D_{N}$ be the finite-dimensional vector space spanned by monomials of degree $N$ and $O_{N}$ the subspace of $E$ consisting of formal series whose coefficients of degree $< N$ vanish, then $(O_{N})_{N \in N}$ is a filtration:

E \supset O_{0} \supset O_{1} \supset \dots, ⋂_{N} O_{N} = {0} .

This filtration will be used for all formal convergences in this section. We shall also need to discuss the degree in

(x, ξ, ℏ)

only; to this aim, we denote by

O_{N} (x, ξ, ℏ)

the subspace of

E

spanned by monomials

x^{α} ξ^{β} ε^{m} ℏ^{ℓ}

such that

| α | + | β | + 2 ℓ \geq N

Let $d, m \in R$ , the symbol class $S^{d} (m)$ is the set of the smooth functions $a_{ε} (., ℏ) : R^{2 n} \times] 0, 1] \to C$ such that, for all $α \in N^{n}$ ,

| \partial_{(x, y)}^{α} a_{ε} (x, ξ, ℏ) | \leq C_{α} ℏ^{d} {(1 + | x |^{2} + | ξ |^{2})}^{m / 2}

for some constant

C_{α} > 0

, uniformly in

ℏ \in (0, ℏ_{0}]

and

ε \in [- ε_{0}, ε_{0}]

, where

ℏ_{0}

and

ε_{0}

are small enough.

S^{d} (m)

is called the space of symbols of order

d

and degree

m

For $a_{ε} \in S^{d} (m)$ and $u \in C_{0}^{\infty} (R^{2 n})$ , the Weyl quantization of $a_{ε}$ acting on $u$ is given by the oscillatory integral

(O p_{ℏ}^{W} (a_{ε}) u) (x) = \frac{1}{(2 π ℏ)^{n}} \int_{R^{2 n}} e^{(i / ℏ) ⟨ x - y, ξ ⟩} a_{ε} (\frac{x + y}{2}, ξ, ℏ) u (y) | d y d ξ | .

In general,

O p_{ℏ}^{W} (a_{ε})

is an unbounded linear operator on

L^{2} (R^{n})

, and

a_{ε}

is called its Weyl symbol. For example, the Weyl symbol of the harmonic oscillator

{\hat{H}}_{2, 0}

is the polynomial

H_{2, 0} = \sum_{j = 1}^{n} \frac{ω_{j, 0}}{2} (ξ_{j}^{2} + x_{j}^{2}),

(6)

and the Weyl symbol of the operator of multiplication by a function

f (x)

is the function

f

itself or, more precisely, the function

(x, ξ) \mapsto f (x)

In this article, we use the Weyl bracket $[f, g]_{W}$ defined on $E$ by the formal Taylor series of the Weyl symbol of the commutator

O p_{ℏ}^{W} \tilde{f} \circ O p_{ℏ}^{W} \tilde{g} - O p_{ℏ}^{W} \tilde{g} \circ O p_{ℏ}^{W} \tilde{f},

where

\tilde{f}

and

\tilde{g}

are smooth symbols whose formal Taylor series is equal to

f

and

g

, respectively. From now on, when

f = f (x, ξ, ε, ℏ)

is a smooth function, we shall allow us to write

f \in E

to signify that its Taylor series at the origin belongs to

E

The filtration $E$ has a nice behavior with respect to the Weyl bracket: if $N_{1} + N_{2} ⩾ 2$ , then, $ℏ^{- 1} [O_{N_{1}}, O_{N_{2}}]_{W} \subset O_{N_{1} + N_{2} - 2}$ .

The Weyl symbol $H_{2, ε}$ of ${\hat{H}}_{2, ε}$ is:

\begin{aligned} H_{2, ε} & = H_{2, 0} + ε L_{2} + ε^{2} M_{2, ε} + O (ε^{\infty} {‖ x ‖}^{2}) \\ = \sum_{j = 1}^{n} \frac{ω_{j, 0}}{2} (ξ_{j}^{2} + x_{j}^{2}) + ε \sum_{j = 1}^{n} \frac{ω_{j, 1}}{2} (ξ_{j}^{2} + x_{j}^{2}) + ε^{2} M_{2, ε} + O (ε^{\infty} {‖ x ‖}^{2}), \end{aligned}

(7)

with

L_{2} = \sum_{j = 1}^{n} \frac{ω_{j, 1}}{2} (ξ_{j}^{2} + x_{j}^{2}),

where

M_{2, ε}

is the Weyl symbol of

{\hat{M}}_{2, ε}

and

L_{2}

is the Weyl symbol of

{\hat{L}}_{2}

To summarize, we have proven the following.

Proposition 3.1

Let ${\hat{P}}_{ε}$ be the Schrödinger operator defined in (1). Then there exists an explicit unitary operator $U_{ε}$ on $L^{2} (R^{n})$ ( $U_{ε} = Λ_{ε} g_{ε} τ_{ε}$ is composed of a translation, a unitary transform, and a scaling, all in the position variable $x$ ) such that the differential operator $U_{ε} {\hat{P}}_{ε} U_{ε}^{*}$ has the following Weyl symbol:

\begin{aligned} σ_{W} (U_{ε} {\hat{P}}_{ε} U_{ε}^{*}) & = min_{R^{n}} V_{ε} + H_{2, 0} + ε L_{2} + L_{ε}^{'}, \\ = min_{R^{n}} V_{ε} + H_{2, 0} + L_{ε}, \end{aligned}

where

H_{2, 0}

and

L_{2}

are defined in (7),

L_{2} \in D_{2}

L_{ε}^{'} = ε^{2} M_{2, ε} + R_{ε} (x) + O (ε^{\infty} {‖ x ‖}^{2}) \in O_{3}

L_{ε} = ε L_{2} + L_{ε}^{'} \in O_{3}

M_{2, ε}

is the Weyl symbol of

{\hat{M}}_{2, ε}

defined in (5), and

R_{ε} (x)

is a smooth function of order

O (x^{3})

at the origin, uniformly with respect to

ε

Let $A \in E$ , we define the adjoint operator ${ad}_{A} (P) := [A, P]_{W}$ , $P \in E$ . The crucial properties of ${ad}_{H_{2, 0}}$ are given in the next proposition, for more details see Charles and Vũ Ng $ọ$ c (2008).

Proposition 3.2

Let $ω_{0} = (ω_{1, 0}, \dots, ω_{n, 0})$ , and let $z_{j} = x_{j} + i ξ_{j}$ ; then ${ad}_{H_{2, 0}}$ is diagonal on the family ${z^{α} {\bar{z}}^{β}, z \in C^{n}, α, β \in N^{n}}$ of $C [[z, \bar{z}]] = C [[x, ξ]]$ and

{ad}_{H_{2, 0}} (z^{α} {\bar{z}}^{β}) = ℏ ⟨ β - α, ω_{0} ⟩ z^{α} {\bar{z}}^{β} .

We have

D_{N} = \ker (ℏ^{- 1} {ad}_{H_{2, 0}}) \oplus I m (ℏ^{- 1} {ad}_{H_{2, 0}}) .

(8)

We are now ready to state the formal $ε$ -Birkhoff–Gustavson normal form, which is a natural extension of the usual Birkhoff–Gustavson case. Note that we prove here the result for a general perturbation $L_{ε} \in O_{3}$ , which does not need to be equal to the particular form obtained above when one reduces a Schrödinger operator.

Theorem 3.3

Let $H_{2, 0}$ be as in (6), and let $L_{ε} \in O_{3}$ , then there exist $A_{ε} \in O_{3}$ and $K_{ε} \in O_{3}$ such that,

\begin{aligned} e^{i ℏ^{- 1} {ad}_{A_{ε}}} (H_{2, 0} + L_{ε}) = H_{2, 0} + K_{ε}, \\ {[H_{2, 0}, K_{ε}]}_{W} = 0. \end{aligned}

(9)

Moreover, the following properties hold.

$K_{ε}$ is unique and $A_{ε}$ is unique modulo $\ker {ad}_{H_{2, 0}}$ .

if $H_{2, 0}$ and $L_{ε}$ have real coefficients (i.e. belong to $R [[x, ξ, ε, ℏ]]$ ) then $K_{ε} \in R [[x, ξ, ε, ℏ]]$ , and $A_{ε}$ can be chosen to have real coefficients as well.

If $L_{ε} \in ε^{m} C [[x, ξ, ε, ℏ]]$ , for some $m \in N$ , then $K_{ε} \in ε^{m} C [[x, ξ, ε, ℏ]]$ as well.

If $L_{ε} \in ℏ^{ℓ} C [[x, ξ, ε, ℏ]]$ , for some $ℓ \in N$ , then $K_{ε} \in ℏ^{ℓ} C [[x, ξ, ε, ℏ]]$ as well.

If $L_{ε} \in O_{N} (x, ξ, ℏ)$ for some $N \geq 0$ , then $K_{ε} \in O_{N} (x, ξ, ℏ)$ as well.

Notice that the sum:

e^{i ℏ^{- 1} {ad}_{A_{ε}}} (H_{2, 0} + L_{ε}) = \sum_{ℓ = 0}^{\infty} \frac{1}{ℓ!} {(\frac{i}{ℏ} {ad}_{A_{ε}})}^{ℓ} (H_{2, 0} + L_{ε}),

is usually not convergent in the analytic sense, even if

P_{ε}

is analytic, but it is always convergent in the formal topology of

E

, because the map

B \mapsto \frac{i}{ℏ} {ad}_{A} (B) = \frac{i}{ℏ} [A, B]_{W}

sends

O_{N}

into

O_{N + 1}

Proof of the theorem.

We construct $A_{ε}$ and $K_{ε}$ inductively, by successive approximations with respect to the filtration $E .$

Let $N \geq 1$ , we suppose that there exists $A_{N, ε} \in O_{3}$ and $K_{N, ε} \in O_{3}$ such:

e^{i ℏ^{- 1} {ad}_{A_{N, ε}}} (H_{2, 0} + L_{ε}) = H_{2, 0} + K_{3, ε} + \dots + K_{N + 1, ε} + R_{N + 2, ε} + O_{N + 3},

(10)

where

K_{j, ε} \in D_{j}

commutes with

H_{2, 0}

and

R_{N + 2, ε} \in D_{N + 2} .

We look for

A_{ε}^{'} \in D_{N + 2}

, and

K_{N + 2, ε} \in D_{N + 2}

that commutes with

H_{2, 0}

, such that:

e^{i ℏ^{- 1} {ad}_{A_{N, ε} + A_{ε}^{'}}} (H_{2, 0} + L_{ε}) = H_{2, 0} + K_{3, ε} + \dots + K_{N + 1, ε} + K_{N + 2, ε} + O_{N + 3} .

This is equivalent to

R_{N + 2, ε} - (i / ℏ) {[H_{2, 0}, A_{ε}^{'}]}_{W} = K_{N + 2, ε} .

Since

H_{2, 0}

satisfies (8), we have

R_{N + 2, ε} = R_{N + 2, ε}^{'} + \frac{i}{ℏ} [H_{2, 0}, R_{N + 2 ε}^{″}]_{W}

, where

R_{N + 2, ε}^{'} \in D_{N + 2}

commutes with

H_{2, 0}

(and is unique) and

R_{N + 2, ε}^{″} \in D_{N + 2}

, unique modulo

\ker {ad}_{H_{2, 0}}

. Thus, we must (and can) choose

K_{N + 2, ε} = R_{N + 2, ε}^{'}

and, we may choose

A_{ε}^{'} = R_{N + 2, ε}^{″}

. This shows that (10) holds at order

N + 1

, proving (9), and the first property.

The second property follows from Moyal’s formula

i [f, g]_{W} (x, ξ, ℏ) = 2 \sin (\frac{ℏ}{2} ◻) (f (x_{1}, ξ_{1}, ℏ) g (x_{2}, ξ_{2}, ℏ)) |_{\binom{x = x_{1} = x_{2},}{ξ = ξ_{1} = ξ_{2}}},

where

◻ = \sum_{j = 1}^{n} \partial_{ξ_{1}^{j}} \partial_{x_{2}^{j}} - \partial_{x_{1}^{j}} \partial_{ξ_{2}^{j}} .

Properties 3 and 4 are due to the fact that

ε

and

ℏ

are central elements in

E

(they commute with everything). The last properties hold because

{ad}_{H_{2, 0}}

preserves the

(x, ξ, ℏ)

-order.

3.2. Normalizing the Near Fermi Resonance

A vector of $n$ frequencies $(ω_{1}, \dots, ω_{n})$ is called resonant if the coefficients $ω_{1}, \dots, ω_{n}$ are linearly dependent over the rationals.

In the extreme case where the rank of these coefficients over $Q$ is one, the frequencies $(ω_{1}, \dots, ω_{n})$ are called completely resonant, and there exist co-prime integers $p_{1}, \dots, p_{n}$ and $λ \in R$ such that $ω_{j} = p_{j} λ$ . In this case, we say that $(ω_{1}, \dots, ω_{n})$ (or the harmonic oscillator $\sum_{j = 1}^{n} \frac{ω_{j}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{j}^{2}} + x_{j}^{2})$ ) exhibits a $p_{1} : \dots : p_{n}$ resonance.

More generally, if there exist an integer $r \in {1, \dots, n}$ , a number $λ \in R$ , and co-prime integers $p_{j_{1}}, \dots, p_{j_{r}}$ such that $ω_{j_{i}} = p_{j_{i}} λ$ , then we say that $(ω_{1}, \dots, ω_{n})$ exhibits a $p_{j_{1}} : \dots : p_{j_{r}}$ resonance.

In this paper, we are interested in near resonances, which we define as follows.

Definition 3.4
Let $ω_{ε} = (ω_{1, ε}, \dots, ω_{n, ε})$ be the frequencies of the harmonic oscillator ${\hat{H}}_{2, ε}$ . One says that $ω_{ε}$ admits a near resonance of type $p_{j_{1}} \approx \dots \approx p_{j_{r}}$ if the map $ε \to ω_{ε}$ is $C^{\infty}$ and $ω_{0}$ admits a resonance of type $p_{j_{1}} : \dots : p_{j_{r}}$ .

For example, we say that there is a near resonance relation of the form $p \approx q$ , where $p, q \in N^{}$ , if there exist $j_{0}, k_{0} \in {1, \dots, n}$ such that $q ω_{j_{0}, 0} = p ω_{k_{0}, 0}$ .

This concept of near resonance was introduced in molecular spectroscopy, where such a phenomenon is extremely common among small molecules, see Joyeux (1998).

In order to better understand the non-quadratic terms $K_{j, ε}$ obtained in Theorem 0.5 from the Hamiltonian ${\hat{P}}_{ε}$ in near resonance, we suggest in this paper to study the typical case of the near Fermi resonance in dimension $2$ , denoted by $1 \approx 2$ (i.e. $ω_{2, 0} = 2 ω_{1, 0}$ ), and we will give an explicit computation of the first nontrivial term of the $ε$ -Birkhoff normal form. Physically, the near Fermi resonance is known to affect the spectroscopy of many molecules, among which we find famous ones such as carbon dioxide ${C O}_{2}$ and carbon disulfide ${C S}_{2}$ , see Joyeux (1996). The physics or chemical literature on Fermi resonance is enormous; interestingly, in a very recent work, the $1 \approx 2$ resonance of the ${C O}_{2}$ molecule was demonstrated to have a direct effect on global warming (Wordsworth et al., 2024). From the point of view of classical mechanics, the Fermi resonance and its bifurcations have been extensively studied, see for instance (Cushman et al., 2007; Hanßmann et al., 2020) and references therein.

Consider on $L^{2} (R^{2})$ the semiclassical Schrödinger operator which is transformed according to Proposition 3.1 into a perturbation of the harmonic oscillator ${\hat{H}}_{2, 0}$ , which we denote by ${\hat{P}}_{ε} (ℏ)$ again:
${\hat{P}}_{ε} (ℏ) = W_{ε} (0) + {\hat{H}}_{2, 0} + ε {\hat{L}}_{2} + ε^{2} {\hat{M}}_{2, ε} + R_{ε} (x) + O (ε^{\infty} {‖ x ‖}^{2}),$
where
$\begin{aligned} {\hat{H}}_{2, 0} & = \frac{ω_{1, 0}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{1}^{2}} + x_{1}^{2}) + \frac{ω_{2, 0}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{2}^{2}} + x_{2}^{2}), \\ {\hat{L}}_{2} & = \frac{ω_{1, 1}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{1}^{2}} + x_{1}^{2}) + \frac{ω_{2, 1}}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{2}^{2}} + x_{2}^{2}), \\ {\hat{M}}_{2, ε} & = \frac{{\tilde{ω}}_{1, 2} (ε)}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{1}^{2}} + x_{1}^{2}) + \frac{{\tilde{ω}}_{2, 2} (ε)}{2} (- ℏ^{2} \frac{\partial^{2}}{\partial x_{2}^{2}} + x_{2}^{2}), \end{aligned}$
and $R_{ε} (x) = O (x^{3})$ at the origin. The associated symbol is
$P_{ε} = W_{ε} (0) + H_{2, 0} + ε L_{2} + L_{ε}^{'},$
where for $z_{j} = x_{j} + i ξ_{j}, j = 1, 2$ , $H_{2, 0} = (1 / 2) (ω_{1, 0} | z_{1} |^{2} + ω_{2, 0} | z_{2} |^{2}), L_{2} = (1 / 2) (ω_{1, 1} | z_{1} |^{2} + ω_{2, 1} | z_{2} |^{2}), L_{ε}^{'} = ε^{2} M_{2, ε} + R_{ε} (x) + O (ε^{\infty}), and M_{2, ε} = (1 / 2) ({\tilde{ω}}_{1, 2} (ε) | z_{1} |^{2} + {\tilde{ω}}_{2, 2} (ε) | z_{1} |^{2}) .$ We now consider the associated Taylor series (which we denote by the same symbols). Thus, $L_{ε}^{'} \in O_{3}$ , and notice that $ε L_{2} \in D_{3}$ and commutes with $H_{2, 0}$ . By applying Theorem 3.3, there exist $A_{1, ε} \in O_{3}$ and $K_{3, ε} \in D_{3}$ such that $K_{3, ε} \in O_{2} (x, ξ, ℏ)$ and
$e^{i ℏ^{- 1} {ad}_{A_{1, ε}}} (H_{2, 0} + ε L_{2} + L_{ε}^{'}) = H_{2, 0} + K_{3, ε} + O_{4},$
hence
$ε L_{2} + L_{ε}^{'} + \frac{i}{ℏ} {[A_{1, ε}, H_{2, 0} + ε L_{2} + L_{ε}^{'}]}_{W} - K_{3, ε} \in O_{4} .$
Let us write the expansion
$R_{ε} (x) = R_{0} (x) + ε R_{1} (x) + ε^{2} R_{2} (x) + \dots,$
where $R_{j} (x) = O (x^{3})$ , and we can write $R_{0} (x) = R_{3, 0} + O (x^{4})$ with $R_{3, 0} \in D_{3}$ . Since $ε^{2} M_{2, ε} \in O_{4}$ , we have $L_{ε}^{'} = R_{3, 0} + O (4)$ . We obtain:
$ε L_{2} + R_{3, 0} = \frac{i}{ℏ} {[H_{2, 0}, A_{1, ε}]}_{W} + K_{3, ε} + O_{4} .$
Since $R_{3, 0} \in D_{3} = \ker (i ℏ^{- 1} {ad}_{H_{2, 0}}) \oplus I m (i ℏ^{- 1} {ad}_{H_{2, 0}})$ , we can write
$R_{3, 0} = {\tilde{R}}_{3, 0} + \frac{i}{ℏ} {[H_{2, 0}, {\tilde{\tilde{R}}}_{3, 0}]}_{W},$
(11)
where ${\tilde{R}}_{3, 0} \in D_{3}$ and commutes with $H_{2, 0}$ , and ${\tilde{\tilde{R}}}_{3, 0} \in D_{3}$ . Consequently,
$K_{3, ε} = ε L_{2} + {\tilde{R}}_{3, 0} .$
Notice that, because $H_{2, 0}$ is quadratic, equation (11) is equivalent to $R_{3, 0} = {\tilde{R}}_{3, 0} + {H_{2, 0}, {\tilde{\tilde{R}}}_{3, 0}}$ ; therefore, ${\tilde{R}}_{3, 0}$ does not depend on $ℏ$ nor $ε$ . Hence, in order to make the condition that ${\tilde{R}}_{3, 0}$ commutes with $H_{2, 0}$ explicit, we may use the known results from the classical normal form, see for instance (Cushman et al., 2007). Let us recall the arguments. By Proposition 3.2, we have
${\tilde{R}}_{3, 0} = \sum_{\binom{| α | + | β | = 3}{⟨ ω_{0}, β - α ⟩ = 0}} c_{α β}^{(3)} z^{α} {\bar{z}}^{β} .$
According to Definition 3.4, a near Fermi resonance for $ω_{ε} = (ω_{1, ε}, ω_{2, ε})$ means an exact Fermi resonance for $ω_{0} = (ω_{1, 0}, ω_{2, 0})$ , that is:
$ω_{2, 0} = 2 ω_{1, 0} .$
Therefore,
$⟨ ω_{0}, β - α ⟩ = 0 \Leftrightarrow (α_{1} + 2 α_{2}) = (β_{1} + 2 β_{2}),$
(12)
where $α = (α_{1}, α_{2}) \in N^{2}$ and $β = (β_{1}, β_{2}) \in N^{2} .$ Now, to obtain ${\tilde{R}}_{3, 0}$ , it is necessary to look for all monomials of order $3$ that satisfy the Fermi resonance relation (12). Thus, ${\tilde{R}}_{3, 0}$ is generated by the monomials ${\bar{z}}_{1}^{2} z_{2}, z_{1}^{2} {\bar{z}}_{2}$ . Since ${\tilde{R}}_{3, 0}$ is real,
$\begin{aligned} {\tilde{R}}_{3, 0} & = μ R e (z_{1}^{2} {\bar{z}}_{2}) + ν I m (z_{1}^{2} {\bar{z}}_{2}) \\ = \frac{μ}{2} ({\bar{z}}_{1}^{2} z_{2} + z_{1}^{2} {\bar{z}}_{2}) + \frac{ν}{2 i} ({\bar{z}}_{1}^{2} z_{2} - z_{1}^{2} {\bar{z}}_{2}), μ, ν \in R . \end{aligned}$
Determination of the coefficients $μ$ and $ν$ : We Taylor expand $R_{3, 0}$ at the origin:
$\begin{aligned} R_{3, 0} (x_{1}, x_{2}) & = \frac{1}{3!} [\frac{\partial^{3} R_{3, 0} (0)}{\partial x_{1}^{3}} x_{1}^{3} + 3 \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{1}^{2} \partial x_{2}} x_{1}^{2} x_{2} \\ + 3 \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{2}^{2} \partial x_{1}} x_{2}^{2} x_{1} + \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{2}^{3}} x_{2}^{3}] . \end{aligned}$
Recall that, due to (8), ${\tilde{R}}_{3, 0}$ is obtained by writing $R_{3, 0}$ in the $(z^{α} {\bar{z}}^{β})$ basis and keeping only the resonant coefficients, that is, the coefficients of $z_{1}^{2} {\bar{z}}_{2}$ and ${\bar{z}}_{1}^{2} z_{2}$ . Since $x_{j} = (1 / 2) (z_{j} + {\bar{z}}_{j})$ , we remark first that only the coefficient of $x_{1}^{2} x_{2}$ in $R_{3, 0} (x_{1}, x_{2})$ contributes to ${\tilde{R}}_{3, 0}$ (by homogeneity considerations), and second that the expansion of $R_{3, 0}$ in the $(z^{α} {\bar{z}}^{β})$ basis has only real* coefficients (which would not be the case for a general Hamiltonian depending also on $ξ$ ). More precisely, we have
$x_{1}^{2} x_{2} = \frac{1}{8} (z_{1}^{2} z_{2} + z_{1}^{2} {\bar{z}}_{2} + {\bar{z}}_{1}^{2} z_{2} + {\bar{z}}_{1}^{2} {\bar{z}}_{2} + 2 | z_{1} |^{2} z_{2} + 2 | z_{1} |^{2} {\bar{z}}_{2}),$
which gives
$\begin{aligned} {\tilde{R}}_{3, 0} & = \frac{1}{16} \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{1}^{2} \partial x_{2}} (z_{1}^{2} {\bar{z}}_{2} + {\bar{z}}_{1}^{2} z_{2}) \\ = \frac{1}{8} \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{1}^{2} \partial x_{2}} R e (z_{1}^{2} {\bar{z}}_{2}) . \end{aligned}$
So, $ν = 0$ , and we obtain the following result.
Theorem 3.5
In dimension $2$ , the quantum $ε$ -Birkhoff-Gustavson normal form of the operator $H_{2, ε} + R_{ε} (x)$ in near Fermi resonance $1 \approx 2$ , is equal to $H_{2, 0} + K_{3, ε} + O_{4}$ with
$K_{3, ε} = \frac{ω_{1, 1}}{2} ε | z_{1} |^{2} + \frac{ω_{2, 1}}{2} ε | z_{2} |^{2} + μ R e (z_{1}^{2} {\bar{z}}_{2}),$
where
$μ = \frac{1}{8} \frac{\partial^{3} R_{3, 0} (0)}{\partial x_{1}^{2} \partial x_{2}} .$
(13)
Moreover, the remainder in $O_{4}$ also belongs to $O_{2} (x, ξ, ℏ)$ .

In order to obtain a fully satisfactory result for the original Schrödinger operator (1), let us now express $μ$ in terms of the potential $V_{0}$ .
Theorem 3.6
Expressed in terms of the original potential $V_{0}$ , the coefficient $μ$ of Theorem 3.5 is given as follows.
$\begin{aligned} μ & = \frac{1}{8 \sqrt{2} ω_{1, 0}^{3 / 2} (1 + z^{2})^{\frac{3}{2}}} (z \partial_{x_{1}^{3}}^{3} V_{0} (0) + (1 - 2 z^{2}) \partial_{x_{1}^{2} x_{2}}^{3} V_{0} (0) \\ + (z^{3} - 2 z) \partial_{x_{1} x_{2}^{2}}^{3} V_{0} (0) + z^{2} \partial_{x_{2}^{3}}^{3} V_{0} (0)) \end{aligned}$
with
$ω_{1, 0} = \frac{1}{\sqrt{3}} {[{(\partial_{x_{1}^{2}}^{2} V_{0} (0) - \partial_{x_{2}^{2}}^{2} V_{0} (0))}^{2} + 4 {(\partial_{x_{1} x_{2}}^{2} V_{0} (0))}^{2}]}^{1 / 4},$
and
$z = \frac{2 | \partial_{x_{1} x_{2}}^{2} V_{0} (0) |}{3 ω_{1, 0}^{2} - \partial_{x_{1}^{2}}^{2} V_{0} (0) + \partial_{x_{2}^{2}}^{2} V_{0} (0)} .$

Proof.
According to Proposition 3.1, see also (4), we can write
$(Λ_{ε} g_{ε} τ_{ε} V_{ε}) (\tilde{y}) = V_{ε} (x_{ε}) + \sum_{j = 1}^{n} \frac{ω_{j, ε}}{2} {\tilde{y}}_{j}^{2} + R_{ε} (\tilde{y}),$
where the variable $\tilde{y}$ is given by the three successive changes of coordinates:
$\tilde{y} = Λ_{ε} U_{ε}^{*} (x - x_{ε}),$
where we abuse the notation $Λ_{ε} (y_{1}, \dots, y_{n}) = (\sqrt{ω_{1, ε}} y_{1}, \dots, \sqrt{ω_{n, ε}} y_{n})$ . Therefore, for all $| α | \geq 3$ ,
$\partial_{\tilde{y}}^{α} R_{ε} (\tilde{y}) = \partial_{\tilde{y}}^{α} (Λ_{ε} g_{ε} τ_{ε} V_{ε}) (\tilde{y}) .$
Since $R_{ε} (\tilde{y}) = R_{3, 0} (\tilde{y}) + O_{4}$ (in the sense of the Taylor series at the origin)
$\frac{\partial^{3} R_{3, 0}}{\partial {\tilde{y}}_{1}^{2} \partial {\tilde{y}}_{2}} (0) = \frac{\partial^{3} R_{0}}{\partial {\tilde{y}}_{1}^{2} \partial {\tilde{y}}_{2}} (0) + O_{4} = \frac{\partial^{3} (Λ_{0} g_{0} τ_{0} V_{0})}{\partial {\tilde{y}}_{1}^{2} \partial {\tilde{y}}_{2}} (0) + O_{4},$
and hence the equality holds without the $O_{4}$ , since the quantities do not depend on $x, ξ, ε, ℏ$ . Moreover, $τ_{0} = I d$ , so
$\frac{\partial^{3} R_{3, 0}}{\partial {\tilde{y}}_{1}^{2} \partial {\tilde{y}}_{2}} (0) = \frac{\partial^{3} (Λ_{0} g_{0} V_{0})}{\partial {\tilde{y}}_{1}^{2} \partial {\tilde{y}}_{2}} (0) = \frac{1}{ω_{1, 0} \sqrt{ω_{2, 0}}} \frac{\partial^{3} (g_{0} V_{0})}{\partial y_{1}^{2} \partial y_{2}} (0) .$
(14)
Let us now compute the derivatives of $(g_{0} V_{0}) (y) = V_{0} (U_{0} y)$ . Let $(a, b)$ and $(c, d)$ be an orthonormal basis of eigenvectors of the Hessian matrix $V_{0}^{″} (0)$ for the eigenvalues $(ω_{1, 0}^{2}, ω_{2, 0}^{2})$ , so that we can take
$U_{0} = (\begin{matrix} a & c \\ b & d \end{matrix}) .$
We have
$\begin{aligned} \frac{\partial^{3} (V_{0} (U_{0} y))}{\partial y_{1}^{2} \partial y_{2}} (0) & = a^{2} c \partial_{x_{1}^{3}}^{3} V_{0} (0) + (2 a b c + a^{2} d) \partial_{x_{1}^{2} x_{2}}^{3} V_{0} (0) \\ + (b^{2} c + 2 a b d) \partial_{x_{1} x_{2}^{2}}^{3} V_{0} (0) + b^{2} d \partial_{x_{2}^{3}}^{3} V_{0} (0) . \end{aligned}$
(15)
It remains to express the coefficients $a, b, c, d$ , taking into account the fact that the eigenvalues of the Hessian matrix ${V^{″}}_{0} (0)$ are $ω_{1, 0}^{2} = λ$ , $ω_{2, 0}^{2} = 4 λ$ ; one can check that this implies
$λ := \frac{1}{3} \sqrt{{(\partial_{x_{1}^{2}}^{2} V_{0} (0) - \partial_{x_{2}^{2}}^{2} V_{0} (0))}^{2} + 4 (\partial_{x_{1} x_{2}}^{2} V_{0} (0))^{2}} .$
In fact, it is elementary to check that for a real symmetric positive matrix whose eigenvalues are $(λ, 4 λ)$ , the discriminant of the characteristic polynomial is equal to $9 λ^{2}$ , and the matrix takes the form, for some $x \in R$ ,
$(\begin{matrix} \frac{x + 5 λ}{2} & \frac{\sqrt{9 λ^{2} - x^{2}}}{2} \\ \frac{\sqrt{9 λ^{2} - x^{2}}}{2} & \frac{5 λ - x}{2} \end{matrix}) .$
Its eigenvectors can be written as
$(1, - z) and (z, 1),$
with $z = \sqrt{(9 λ^{2} - x^{2})} / (3 λ - x) \geq 0$ . Upon normalization, we obtain
$\begin{aligned} a = \frac{1}{\sqrt{1 + z^{2}}} b = \frac{- z}{\sqrt{1 + z^{2}}} \\ c = \frac{z}{\sqrt{1 + z^{2}}} d = \frac{1}{\sqrt{1 + z^{2}}} . \end{aligned}$
In our case, we have
$x = \partial_{x_{1}^{2}}^{2} V_{0} (0) - \partial_{x_{2}^{2}}^{2} V_{0} (0),$
therefore
$\begin{aligned} z & = \frac{2 | \partial_{x_{1} x_{2}}^{2} V_{0} (0) |}{3 λ - x} \\ = \frac{2 | \partial_{x_{1} x_{2}}^{2} V_{0} (0) |}{{({(\partial_{x_{1}^{2}}^{2} V_{0} (0) - \partial_{x_{2}^{2}}^{2} V_{0} (0))}^{2} + 4 {(\partial_{x_{1} x_{2}}^{2} V_{0} (0))}^{2})}^{1 / 2} - \partial_{x_{1}^{2}}^{2} V_{0} (0) + \partial_{x_{2}^{2}}^{2} V_{0} (0)}, \end{aligned}$
and
$\begin{aligned} a^{2} c & = \frac{z}{(1 + z^{2})^{3 / 2}} \\ 2 a b c + a^{2} d & = \frac{1 - 2 z^{2}}{(1 + z^{2})^{3 / 2}} \\ b^{2} c + 2 a b d & = \frac{z^{3} - 2 z}{(1 + z^{2})^{3 / 2}} \\ b^{2} d & = \frac{z^{2}}{(1 + z^{2})^{3 / 2}}, \end{aligned}$
which, in view of (13), (14), and (15), proves the theorem.
4. Spectral Analysis in Near Fermi Resonance

Theorem 3.3 gives a formal conjugation of the initial Schrödinger Hamiltonian ${\hat{P}}_{ε}$ into an operator of the form $c_{ε} + {\hat{H}}_{2, 0} + {\hat{K}}_{ε}$ , where $c_{ε} = W_{ε} (0) = min V_{ε}$ , see Proposition 3.1. Therefore, it is expected that, in regimes where $E$ and $ℏ$ are small enough, the spectrum inside $(- \infty, E)$ of the normal form $c_{ε} + {\hat{H}}_{2, 0} + {\hat{K}}_{ε}$ , restricted to the spectral subspace of ${\hat{H}}_{2, 0}$ corresponding to energies in $[0, (1 + η) E]$ , where $η > 0$ is fixed, is a good approximation of the spectrum of ${\hat{P}}_{ε}$ in the interval $(- \infty, E)$ , see Charles and Vũ Ng $ọ$ c (2008).

The goal of this section is to describe the spectrum of the $ε$ -Birkhoff–Gustavson normal form of ${\hat{P}}_{ε}$ in the case of a near Fermi resonance, in terms of the original potential $V_{ε}$ . We shall use the expression of the normal form modulo $O_{4}$ given in the previous section. Since the remainder also belongs to $O_{2} (x, ξ, ℏ)$ by Theorem 3.5, it belongs to

O_{4} (x, ξ, ℏ) + ε O_{3} (x, ξ, ℏ) + ε^{2} O_{2} (x, ξ, ℏ) .

Therefore, using Charles and Vũ Ng

ọ

c (2008: Lemma 4.2), the normal form at order 3, that is to say

c_{ε} + {\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}

, is expected to approximate the spectrum of

{\hat{P}}_{ε}

below some energy

E

with a precision of order

O (E^{2}) + O (ε E^{3 / 2}) + O (ε^{2} E)

. For instance, if we are interested in a fixed number of low-lying eigenvalues, we may take

E = C ℏ

for some constant

C > 0

and we obtain a precision of order

O (ℏ^{2}) + O (ε ℏ^{3 / 2}) + O (ε^{2} ℏ)

. In order to compute the matrix elements of the normal form, we shall need to understand the Weyl quantization

{\hat{K}}_{3, ε}

K_{3, ε}

, which is

\begin{aligned} {\hat{K}}_{3, ε} & = ε {\hat{L}}_{2} + μ \hat{R e (z_{1}^{2} {\bar{z}}_{2})}, \\ = ε {\hat{L}}_{2} + μ (x_{1}^{2} x_{2} - 2 ℏ^{2} x_{1} \frac{\partial}{\partial x_{1}} \frac{\partial}{\partial x_{2}} + ℏ^{2} x_{2} \frac{\partial^{2}}{\partial x_{1}^{2}} - ℏ^{2} \frac{\partial}{\partial x_{2}}) . \end{aligned}

For that purpose, it will be convenient to pass to the Bargmann representation.

4.1. Creation and Annihilation Operators

Let us introduce

\begin{matrix} a_{j} (ℏ) = \frac{1}{\sqrt{2 ℏ}} (x_{j} + ℏ \frac{\partial}{\partial x_{j}}); \\ b_{j} (ℏ) = \frac{1}{\sqrt{2 ℏ}} (x_{j} - ℏ \frac{\partial}{\partial x_{j}}), \end{matrix}

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which are, respectively, called the operators of annihilation and creation, acting as unbounded operators on

L^{2} (R^{n})

(see, for instance, Combescure & Robert, 2012).

Formally, the operators $a_{j} (h)$ and $b_{j} (h)$ satisfy the following properties:

\begin{array}{cll} a_{j}^{*} (ℏ) = b_{j} (ℏ), & b_{j}^{*} (ℏ) = a_{j} (ℏ), \\ [a_{j} (ℏ), b_{k} (ℏ)] = δ_{j k}, & [a_{j} (ℏ), a_{k} (ℏ)] = 0, & [b_{j} (ℏ), b_{k} (ℏ)] = 0. \end{array}

Using

a_{j} (ℏ)

and

b_{j} (ℏ)

to rewrite

{\hat{H}}_{2, 0}

, we have

\begin{aligned} {\hat{H}}_{2, 0} = ℏ (ω_{1, 0} (a_{1} (ℏ) b_{1} (ℏ) - \frac{1}{2}) + ω_{2, 0} (a_{2} (ℏ) b_{2} (ℏ) - \frac{1}{2})), \\ {\hat{L}}_{2} = ℏ (ω_{1, 1} (a_{1} (ℏ) b_{1} (ℏ) - \frac{1}{2}) + ω_{2, 1} (a_{2} (ℏ) b_{2} (ℏ) - \frac{1}{2})), \end{aligned}

and

{\hat{K}}_{3, ε}

is given by

\begin{aligned} {\hat{K}}_{3, ε} & = ℏ ε (ω_{1, 1} (a_{1} (ℏ) b_{1} (ℏ) - \frac{1}{2}) + ω_{2, 1} (a_{2} (ℏ) b_{2} (ℏ) - \frac{1}{2})) \\ + \sqrt{2} μ ℏ^{3 / 2} (a_{2} (ℏ) b_{1}^{2} (ℏ) + a_{1}^{2} (ℏ) b_{2} (ℏ)) . \end{aligned}

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4.2. Bargmann Representation

In this section, we review several fundamental results relating to the Bargmann transform and the space $B_{F}$ of Bargmann–Fock, also known as the Bargmann space (Bargmann, 1961). For further details see Combescure and Robert (2012). Let us consider the space

B_{F} = {φ (ζ) holomorphic function on C^{n}; \int_{C^{n}} | φ (ζ) |^{2} d μ_{n} (ζ) < + \infty},

where

d μ_{n} (ζ)

is the Gaussian measure defined by

d μ_{n} (ζ) = e^{- | ζ |^{2} / ℏ} d^{n} ζ

ζ_{j} = (x_{j} - i ξ_{j}) / \sqrt{2} = \bar{z_{j}} / \sqrt{2}

, and

d^{n} ζ

is the Lebesgue measure in

(x, ξ)

B_{F}

is a Hilbert space equipped with a natural inner product

⟨ f, g ⟩ = \int_{C^{n}} f (ζ) \bar{g (ζ)} d μ_{n} (ζ) .

The semiclassical scaling that we use here can be found, for instance, in Combescure and Robert (2012: Proposition 7).

Theorem 4.1 Bargmann, 1961

There is a unitary mapping $T_{B}$ from $L^{2} (R^{n})$ to $B_{F}$ defined by

(T_{B} f) (ζ) = \frac{1}{2^{n / 2} (π ℏ)^{3 n / 4}} \int_{R^{n}} f (x) e^{- (1 / ℏ) [(1 / 2) (ζ^{2} + x^{2}) + \sqrt{2} x ζ]} d^{n} x,

T_{B}

is known as the Bargmann transform.

The Bargmann representation is particularly suited for studying harmonic approximations, as creation and annihilation operators become expressible in a very simple way.

Proposition 4.1 Bargmann, 1961

We have

T_{B} (a_{j} (ℏ)) {T_{B}}^{- 1} = \frac{\partial}{\partial ζ_{j}} and T_{B} (b_{j} (ℏ)) {T_{B}}^{- 1} = ζ_{j},

where

ζ_{j}

represents the operator of multiplication by

ζ_{j}

Thus, the harmonic oscillator’s Bargmann transform is,

{\hat{H}}_{2, 0}^{B} := T_{B} ({\hat{H}}_{2, 0}) {T_{B}}^{- 1} = ℏ \sum_{j = 1}^{n} ω_{j, 0} (ζ_{j} \frac{\partial}{\partial ζ_{j}} + \frac{1}{2}) .

It is then standard to prove that the functions

{φ_{α} := \frac{ζ^{α}}{(2 π ℏ)^{n / 2} \sqrt{α!}}}_{α \in N^{n}}

form an orthonormal Hilbertian basis of

B_{F}

, on which

{\hat{H}}_{2, 0}^{B}

is diagonal. It follows that

{\hat{H}}_{2, 0}^{B}

is essentially self-adjoint with a discrete spectrum which consists of the eigenvalues

λ_{α}

α \in N^{n}

given by:

σ ({\hat{H}}_{2, 0}^{B}) = {λ_{α} = ℏ (u + \frac{| ω_{0} |}{2}), u = ⟨ ω_{0}, α ⟩ = \sum_{j = 1}^{n} ω_{j, 0} α_{j}, α \in N^{n}},

(repeated with their multiplicities) where

ω_{0} = (ω_{1, 0}, \dots, ω_{n, 0})

α = (α_{1}, \dots, α_{n})

and

| ω_{0} | = \sum_{j = 1}^{n} ω_{j, 0}

. The corresponding eigenspaces are

H_{u}^{B} = S p a n {\frac{ζ^{α}}{(2 π ℏ)^{n / 2} \sqrt{α!}}, ⟨ ω_{0}, α ⟩ = u} .

4.3. Spectrum in Near Fermi Resonance

Charles and Vũ Ng $ọ$ c (2008) give a detailed study of the spectrum of semiclassical pseudodifferential operators whose harmonic approximation possesses an exact complete resonance; the idea was to restrict the Birkhoff normal form to the eigenspaces of the resonant harmonic oscillator. In this section, we extend this result to the case of a near Fermi resonance, using the $ε$ -Birkhoff–Gustavson normal form $H_{2, 0} + K_{3, ε} + O_{4}$ of Theorem 3.3.

First, let us rewrite this normal form in the Bargmann representation. According to (17), we have

\begin{array}{cll} {\hat{K}}_{3, ε}^{B} & := & T_{B} ({\hat{K}}_{3, ε}) {T_{B}}^{- 1} = ℏ ε ω_{1, 1} (ζ_{1} \frac{\partial}{\partial ζ_{1}} + \frac{1}{2}) + ℏ ε ω_{2, 1} (ζ_{2} \frac{\partial}{\partial ζ_{2}} + \frac{1}{2}) \\ + \sqrt{2} μ ℏ^{3 / 2} (ζ_{2} \frac{\partial^{2}}{\partial ζ_{1}^{2}} + ζ_{1}^{2} \frac{\partial}{\partial ζ_{2}}) . \end{array}

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Thus, in order to obtain a good approximation of the spectrum of our Schrödinger operator

{\hat{P}}_{ε}

, we shall compute the spectrum of the restriction of

{\hat{K}}_{3, ε}^{B}

to the eigenspaces of

{\hat{H}}_{2, 0}^{B}

(recall that

{\hat{H}}_{2, 0}^{B}

commutes with

{\hat{K}}_{3, ε}^{B}

). To this aim, we calculate the matrix elements of

{\hat{K}}_{3, ε}^{B}

in the basis

φ_{α}

H_{N}^{B}

, where

N \in N

and

H_{N}^{B} = S p a n {φ_{α} = \frac{ζ^{α_{1}} ζ^{α_{2}}}{2 π ℏ \sqrt{α_{1}! α_{2}!}}; α = (α_{1}, α_{2}) \in N^{2}, α_{1} + 2 α_{2} = N} .

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As in Ghomari et al. (2013), we use

\begin{aligned} \frac{\partial φ_{(α_{1}, α_{2})}}{\partial ζ_{1}} & = \sqrt{α_{1}} φ_{(α_{1} - 1, α_{2})}, \\ \frac{\partial^{2} φ_{(α_{1}, α_{2})}}{\partial ζ_{1}^{2}} & = \sqrt{α_{1}} \sqrt{α_{1} - 1} φ_{(α_{1} - 2, α_{2})}, \\ \frac{\partial φ_{(α_{1}, α_{2})}}{\partial ζ_{2}} & = \sqrt{α_{2}} φ_{(α_{1}, α_{2} - 1)}, \\ \frac{\partial^{2} φ_{(α_{1}, α_{2})}}{\partial ζ_{2}^{2}} & = \sqrt{α_{2}} \sqrt{α_{2} - 1} φ_{(α_{1}, α_{2} - 2)}, \\ \frac{\partial^{2} φ_{(α_{1}, α_{2})}}{\partial ζ_{1} \partial ζ_{2}} & = \sqrt{α_{1}} \sqrt{α_{2}} φ_{(α_{1} - 1, α_{2} - 1)}, \end{aligned}

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and

\begin{aligned} ζ_{1} φ_{(α_{1}, α_{2})} & = \sqrt{α_{1} + 1} φ_{(α_{1} + 1, α_{2})}, \\ ζ_{1}^{2} φ_{(α_{1}, α_{2})} & = \sqrt{α_{1} + 2} \sqrt{α_{1} + 1} φ_{(α_{1} + 2, α_{2})}, \\ ζ_{2} φ_{(α_{1}, α_{2})} & = \sqrt{α_{2} + 1} φ_{(α_{1}, α_{2} + 1)}, \\ ζ_{2}^{2} φ_{(α_{1}, α_{2})} & = \sqrt{α_{2} + 2} \sqrt{α_{2} + 1} φ_{(α_{1}, α_{2} + 2)}, \\ ζ_{1} ζ_{2} φ_{(α_{1}, α_{2})} & = \sqrt{α_{1} + 1} \sqrt{α_{2} + 1} φ_{(α_{1} + 1, α_{2} + 1)} . \end{aligned}

According to (18), we obtain, with

α = (α_{1}, α_{2})

\begin{aligned} {\hat{K}}_{3, ε}^{B} φ_{α} = & ℏ ε ω_{1, 1} (ζ_{1} \partial_{ζ_{1}} φ_{α} + \frac{1}{2} φ_{α}) + ℏ ε ω_{2, 1} (ζ_{2} \partial_{ζ_{2}} φ_{α} + \frac{1}{2} φ_{α}) \\ + \sqrt{2} μ ℏ^{3 / 2} (ζ_{2} \sqrt{α_{1}} \sqrt{α_{1} - 1} φ_{(α_{1} - 2, α_{2})} + ζ_{1}^{2} \sqrt{α_{2}} φ_{(α_{1}, α_{2} - 1)}), \\ = & ℏ (ε ω_{1, 1} (α_{1} + \frac{1}{2}) + ε ω_{2, 1} (α_{2} + \frac{1}{2})) φ_{α} \\ + \sqrt{2} μ ℏ^{3 / 2} (\sqrt{(α_{2} + 1) α_{1} (α_{1} - 1)} φ_{(α_{1} - 2, α_{2} + 1)} \\ + \sqrt{(α_{1} + 2) (α_{1} + 1) α_{2}} φ_{(α_{1} + 2, α_{2} - 1)}) . \end{aligned}

We can confirm on these formulas that the space

H_{N}^{B}

is stable by

{\hat{K}}_{3, ε}^{B}

because:

α_{1} - 2 + 2 (α_{2} + 1) = α_{1} + 2 α_{2} = N and α_{1} + 2 + 2 (α_{2} - 1) = α_{1} + 2 α_{2} = N .

We can also confirm the Hermitian symmetry of the matrix

{\hat{K}}_{3, ε}^{B}

. Indeed, let us order the basis

φ_{α}

according to

ℓ = 0, 1, \dots, E [N / 2]

and

α_{ℓ} = (N - 2 ℓ, ℓ)

. The image of

φ_{α_{ℓ}}

is a vector with three components, on

φ_{α_{ℓ}}

φ_{α_{ℓ - 1}}

, and

φ_{α_{ℓ + 1}}

. Thus, the matrix is symmetric if and only if the coefficient of

{\hat{K}}_{3, ε}^{B} (φ_{α_{ℓ}})

φ_{α_{ℓ + 1}}

is equal to the coefficient of

{\hat{K}}_{3, ε}^{B} (φ_{α_{ℓ + 1}})

φ_{α_{ℓ}}

. The first one is equal to

\sqrt{2} μ ℏ^{3 / 2} \sqrt{(α_{2} + 1) α_{1} (α_{1} - 1)},

with

α_{1} = ℓ

α_{2} = N - 2 ℓ

, and the second one is

\sqrt{2} μ ℏ^{3 / 2} \sqrt{(α_{1}^{'} + 2) (α_{1}^{'} + 1) α_{2}^{'}},

with

α_{1}^{'} = α_{1} - 2

α_{2}^{'} = α_{2} + 1

, so the equality of these coefficients is clear.

In matrix form, we obtain

Theorem 4.2

For any $N \geq 0$ , in the basis $(φ_{α})$ of the space $H_{N}^{B}$ given in (19), ordered by $α_{2}$ , of size $⌊ N / 2 ⌋ + 1$ , the matrix of the operator ${\hat{K}}_{3, ε}^{B}$ is

M_{ε, N} = (\begin{matrix} d_{N, 0} & A_{N, 0} \\ A_{N, 0} & d_{N, 1} & ⋱ & 0 \\ ⋱ & ⋱ & ⋱ \\ A_{N, ℓ - 1} & d_{N, ℓ} & A_{N, ℓ} \\ 0 & A_{N, ℓ} & d_{N, ℓ + 1} & ⋱ \\ A_{N, ℓ + 1} & ⋱ & ⋱ \\ ⋱ & ⋱ \end{matrix}),

where for

ℓ = 0, 1, \dots, ⌊ N / 2 ⌋

{\begin{aligned} A_{N, ℓ} = & \sqrt{2} μ ℏ^{3 / 2} \sqrt{(ℓ + 1) (N - 2 ℓ) (N - 2 ℓ - 1)} \\ d_{N, ℓ} = & ℏ ε ω_{1, 1} (N - 2 ℓ + \frac{1}{2}) + ℏ ε ω_{2, 1} (ℓ + \frac{1}{2}), \end{aligned}

and

μ

is defined in Theorem 3.6.

Finally, note that the spectrum of ${\hat{H}}_{2, 0}^{B} + {\hat{K}}_{3, ε}^{B}$ restricted to $H_{N}^{B}$ is exactly

⋃_{N \in N} {ℏ ω_{1, 0} (N + \frac{3}{2}) + μ_{ε, N, j}; j = 0, \dots ⌊ \frac{N}{2} ⌋},

where

(μ_{ε, N, 0}, \dots, μ_{ε, N, ⌊ N / 2 ⌋})

are the eigenvalues of the matrix

M_{ε, N}

of Theorem 4.2.

5. Numerical Simulations for the Near Fermi Resonance

We now illustrate our results numerically for the following family of potentials:

V_{ε} (x_{1}, x_{2}) = (\frac{1}{2} + ε) x_{1}^{2} + (2 + 2 c ε) x_{2}^{2} + x_{1}^{3} + \frac{1}{2} x_{1} x_{2}^{2} + γ x_{1}^{2} x_{2} + x_{1}^{4} + x_{2}^{4},

where

c \in R

and

γ

are fixed, and

ε \in R

is such that

\frac{1}{2} + ε > 0 and 1 + c ε > 0 .

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In view of the normalization (3), this potential corresponds to

W_{ε} (0) = 0

ω_{1, ε} := \sqrt{1 + 2 ε}

and

ω_{2, ε} = 2 \sqrt{1 + c ϵ}

. Therefore,

ω_{1, 0} = 1, ω_{2, 0} = 2, ω_{1, 1} = 1, ω_{2, 1} = c .

Since the quadratic part of the potential is already diagonalized, we can take

U = I d

, and hence it is easy to compute

z = 0

and

μ = \frac{γ}{4 \sqrt{2}}

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from Theorem 0.8.

Due to the term $x_{1}^{4} + x_{2}^{4}$ , the potential $V_{ε}$ is confining, and hence ${\hat{P}}_{ε} := - (ℏ^{2} / 2) Δ + V_{ε}$ has a discrete spectrum, bounded from below, and whose eigenvalues form a non-decreasing sequence tending to $+ \infty$ . We are interested in the eigenvalues that belong to the interval $(- \infty, E)$ , where $E \to 0$ as $ℏ \to 0$ .

5.1. Numerical Computation of the Spectrum of

{\hat{P}}_{ε}

In order to numerically compute the spectrum of ${\hat{P}}_{ε}$ without any a priori, we use a general spectral method, not specially adapted to the $1 : 2$ resonance. Namely, we consider the Hermite functions $H_{α_{1}, α_{2}} (x_{1}, x_{2})$ , which correspond via the Bargmann transform to the functions $φ_{α_{1}, α_{2}}$ from (19). We order them according, first, to the energy level $ℏ (α_{1} + α_{2} + 1)$ of the usual harmonic oscillator $- (ℏ^{2} / 2) Δ + (1 / 2) (x_{1}^{2} + x_{2}^{2})$ , and then according to $α_{1}$ . In other words, the first few pairs $(α_{1}, α_{2})$ in increasing order are:

(0, 0), (0, 1), (1, 0), (0, 2), (1, 1), (2, 0), (0, 3), (1, 2), (2, 1), (3, 0), etc.

The next step of the numerical method is to truncate the basis

(H_{α})

into a finite family adapted to the chosen spectral window; according to the semiclassical theory, for a given

E

, eigenfunctions corresponding to eigenvalues less than

E

must be microlocalized in the phase space region

Ω_{ε}

given by

(1 / 2) {‖ ξ ‖}^{2} + V_{ε} (x) \leq E

. Assume that the parameters

γ, c

are chosen such that

V_{ε}

admits a unique minimum at

x = 0

, for

ε

small enough (see Proposition 5.1). By construction, we have

\frac{1}{2} {‖ ξ ‖}^{2} + V_{ε} (x) = \frac{1}{2} (x_{1}^{2} + ξ_{1}^{2}) + 2 x_{2}^{2} + \frac{1}{2} ξ_{2}^{2} + O (ε x^{2}) + O (x^{3}) .

Therefore, if

E

and

ε

are small enough,

Ω_{ε}

is contained in the ball of radius

\sqrt{3 E}

in phase space. (The number

3

could be replaced by any number larger than

2

. For large

E

, one could do better by using the confinement gained by the quartic term

x^{4}

, but in this work, we are mainly interested in small

E

). Taking into account that a wave function localized inside a ball may extend slightly beyond the ball, typically at a distance of order

O (\sqrt{ℏ})

, we can decide to truncate the basis for

(α_{1}, α_{2})

satisfying

ℏ (α_{1} + α_{2} + 1) \leq (3 / 2) E + \sqrt{ℏ}

; thus we shall take

α_{1} + α_{2} \leq M

with

M = \frac{3 E}{2 ℏ} + \frac{1}{\sqrt{ℏ}} .

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We obtain a basis

B_{M}

of cardinal

(M + 1) (M + 2) / 2

which, when

E

is bounded and

ℏ

is small, is

O (E^{2} / ℏ^{2})

On this basis, the action of differential operators with polynomial coefficients can be explicitly computed, similarly to (20); namely, from (16) and Proposition 4.1, we have

x_{j} H_{α} = \sqrt{\frac{ℏ}{2}} (\sqrt{α_{j}} H_{α - 1_{j}} + \sqrt{α_{j} + 1} H_{α + 1_{j}})

and

ℏ \partial_{x_{j}} H_{α} = \sqrt{\frac{ℏ}{2}} (\sqrt{α_{j}} H_{α - 1_{j}} - \sqrt{α_{j} + 1} H_{α + 1_{j}}),

where we have denoted by

(1_{1}, 1_{2})

the canonical basis of

Z^{2}

. In order to implement the operator of multiplication by

x_{j}^{2}

, instead of writing the explicit formula, we may also simply square the truncated matrix for

x_{j}

; by doing so, of course, we introduce an error due to the fact that matrix truncation does not commute with matrix multiplication. The wrong columns concern the images of

H_{α}

when

α + 1_{j} \notin [0, M] \times [0, M]

, that is,

α_{1} + α_{2} + 1 > M

. In other words, in order to obtain a correct matrix, we need to delete the “highest energy level,” which means truncating the computed matrix to the smaller basis

B_{M - 1}

. Similarly, since the matrix for

x_{j}^{2}

has a band structure of width 5 (instead of 3 for

x_{j}

), when computing

x_{j}^{3}

we need to reduce

M

by 2 more. Finally, the truncated matrix for

x_{j}^{4}

will be exact if we reduce

M

by 3 more. The analogous discussion holds for

ℏ \partial_{x_{j}}

. Consequently, in order to obtain the matrix for

{\hat{P}}_{ε}

B_{M}

, we need to start from the larger basis

B_{M + 6}

In view of the above discussion, we can now implement, explicitly, the matrix of ${\hat{P}}_{ε}$ on the basis $B_{M} = {H_{α}, | α | \leq M}$ , and call a standard diagonalization routine for real symmetric matrices. On a standard laptop, this can be easily done for a matrix of size $1000 \times 1000$ , which means $M \leq 43$ .

Our first experiment is to test the validity of the truncation (23). Taking $ℏ = 0.01$ and $E = 10 ℏ$ , formula (23) gives $M = 25$ , which corresponds to a matrix of size $351 \times 351$ . In Figure 1, we vary $M$ around that value and plot the $ℓ_{\infty}$ norm of the difference between the spectrum below $E$ computed with the given $M$ value (i.e. using the matrix obtained from the basis $B_{M}$ ) and the spectrum computed using the largest $M$ (here $M_{m a x} = 35$ ). This experiment confirms that the value predicted by (23) is large enough to obtain a very good accuracy.

Figure 1.

Quality of the numerical spectrum in terms of $M$ . With $ℏ = 0.01$ , $ε = 0$ , and $E = 10 ℏ$ , the value given by (23) is $M = 25$ . We plot here ${‖ {s p}_{M} - {s p}_{M_{m a x}} ‖}_{\infty}$ with $M_{m a x} = 35$ , and ${s p}_{M}$ is the spectrum below $E$ obtained by diagonalizing the truncated matrix of ${\hat{P}}_{ε}$ in the basis $B_{M}$ . The same data with a $\log_{10}$ scale is plotted on the right picture. We see that, at $M \geq 25$ , the error is indeed negligible. Interestingly, we also see that simply choosing $M = 3 E / 2 ℏ = 15$ would not be sufficient. (a) ${‖ {s p}_{M} - {s p}_{M_{m a x}} ‖}_{\infty}$ ; (b) $\log_{10} ({‖ {s p}_{M} - {s p}_{M_{m a x}} ‖}_{\infty})$ .

In Figure 2, we compare the spectrum of ${\hat{P}}_{ε}$ obtained with the above numerical method to the spectrum of the unperturbed harmonic oscillator $H_{2, 0}$ . Recall that the oscillator spectrum is explicit

s p (H_{2, 0}) = {ℏ ω_{1, 0} (α_{1} + 2 α_{2} + \frac{3}{2}), (α_{1}, α_{2}) \in N^{2}}

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and features the famous phenomemon of clustering of eigenvalues on the ladder

ℏ ω_{1, 0} (N + \frac{3}{2}), N \geq 0

, which correspond to polyads in the chemistry literature. While we can recognize the footprints of these polyads on the spectrum of

{\hat{P}}_{ε}

, we notice that much of the structure is lost, even for relatively small

ε

Figure 2.

Spectrum of ${\hat{P}}_{ε}$ (green boxes, solid line) on top of the spectrum of the $1 : 2$ harmonic oscillator ${\hat{H}}_{2, 0}$ (red discs, dotted line).

5.2.

V_{ε}

has a Global Minimum

Our goal is to compare the spectrum of ${\hat{P}}_{ε}$ with what the $ε$ -Birkhoff–Gustavson normal form gives. In order to apply our results, let us first check that $V_{ε}$ has a unique non-degenerate minimum, when the constants $γ, c, ε$ are properly chosen.

Proposition 5.1
Assume that, in addition to (21), the following conditions hold:
$\begin{aligned} 1 - \frac{γ^{2}}{8 (1 + c ε)} > 0, \\ \frac{7}{16} + ε - \frac{1}{4 (1 - (γ^{2} / 8 (1 + c ε)))} > 0. \end{aligned}$
Then, the potential $V_{ε}$ admits a unique non-degenerate minimum at the origin. In particular, if $| γ | < \sqrt{24 / 7}$ and $| ε |$ is small enough, then $V_{ε}$ admits a unique non-degenerate minimum at the origin.
Proof.
We simply complete three squares: first, we write
$\frac{1}{2} x_{1} x_{2}^{2} + x_{4}^{4} = {(x_{2}^{2} + \frac{x_{1}}{4})}^{2} - \frac{x_{1}^{2}}{16} .$
Then, we use
$(2 + 2 c ε) x_{2}^{2} + γ x_{1}^{2} x_{2} = 2 (1 + c ε) {(x_{2} + \frac{γ x_{1}^{2}}{4 (1 + c ε)})}^{2} - \frac{γ^{2} x_{1}^{4}}{8 (1 + c ε)} .$
Incorporating the last term with the monomial $x_{1}^{4}$ from $V_{ε}$ , we finally write, with $κ := 1 - [γ^{2} / 8 (1 + c ε)]$ ,
$κ x_{1}^{4} + x_{1}^{3} = x_{1}^{2} (κ {(x_{1} + \frac{1}{2 κ})}^{2} - \frac{1}{4 κ}) .$
This finally gives
$\begin{aligned} V_{ε} = & (\frac{1}{2} + ε - \frac{1}{16} - \frac{1}{4 κ}) x_{1}^{2} + 2 (1 + c ε) {(x_{2} + \frac{γ x_{1}^{2}}{4 (1 + c ε)})}^{2} \\ + κ x_{1}^{2} {(x_{1} + \frac{1}{2 κ})}^{2} + {(x_{2}^{2} + \frac{x_{1}}{4})}^{2} . \end{aligned}$
Thus, under the conditions of the proposition, we have a sum of four non-negative terms; the sum vanishes if and only if all terms vanish, which is equivalent to $x_{1} = x_{2} = 0$ .

Now, if $γ$ and $c$ are fixed, the two conditions take the form
$\begin{aligned} 1 - \frac{γ^{2}}{8} + O (ε) > 0 \\ \frac{7}{16} - \frac{1}{4 (1 - \frac{γ^{2}}{8})} + O (ε) > 0 \end{aligned}$
which holds, if $ε$ is small enough, as soon as
$\frac{7}{16} - \frac{1}{4 (1 - \frac{γ^{2}}{8})} > 0, that is, γ^{2} < \frac{24}{7},$
which is stronger than the first condition $γ^{2} < 8$ .
Remark 5.2
We don’t claim that the two conditions of Proposition 5.1 are necessary. But they allow for a simple proof, and they are sufficient for our numerical purposes $γ = 0$ and $γ = 1$ .
5.3. Joint Spectrum of ${\hat{H}}_{2, 0}$ and ${\hat{K}}_{3, ε}$

Let us apply the $ε$ -Birkhoff–Gustavson normal form of Theorem 3.3 to order 3: we conjugate the initial Schrödinger operator ${\hat{P}}_{ε}$ to an operator of the form $H_{2, 0} + K_{3, ε} + O_{4}$ , and our goal is perform numerical computations neglecting the $O_{4}$ term.

The operator ${\hat{K}}_{3, ε}$ can be explicitly implemented thanks to Theorems 3.5, 3.6 and 4.2. For any given $N \geq 0$ , we obtain the exact matrix for ${\hat{K}}_{3, ε}$ in the orthonormal basis (19), of cardinal $⌊ N / 2 ⌋ + 1$ .

In order to obtain an approximation of the spectrum of ${\hat{P}}_{ε}$ , following Section 4, we fix some energy $E > 0$ , and we compute the spectrum of ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ restricted to the spectral subspace of ${\hat{H}}_{2, 0}$ corresponding to energies in $[0, (1 + η) E]$ , where $η > 0$ is fixed. In view of the Bargmann representation of Section 4.2, this is equivalent to computing the spectrum of the restriction of ${\hat{H}}_{2, 0}^{B} + {\hat{K}}_{3, ε}^{B}$ to the space $\oplus_{n = 0}^{N} H_{n}^{B}$ , ee (19), for some $N$ large enough. Specifically, the integer $N$ has to be chosen such that $ℏ (\tilde{n} + (3 / 2)) + σ (({\hat{K}}_{3, ε}^{B})_{↾ H_{\tilde{n}}^{B}})$ does not intersect the interval $(0, E)$ for all $\tilde{n} > N$ .

A nice way of displaying this computation is to plot the joint spectrum of the commuting operators $ℏ^{- 1} {\hat{H}}_{2, 0}$ and ${\hat{K}}_{3, ε}$ : this is the set of pairs $(λ_{1}, λ_{2}) \in R^{2}$ such that, with the notation of Theorem 4.2,

\begin{aligned} λ_{1} & = ω_{1, 0} (N + \frac{3}{2}) \\ λ_{2} & = μ_{ε, N, j} for some j = 0, \dots, ⌊ \frac{N}{2} ⌋, \end{aligned}

see Figure 7. Then, the spectrum of

{\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}

is obtained from the joint spectrum by applying the map

(λ_{1}, λ_{2}) \mapsto ℏ λ_{1} + λ_{2}

Notice that the computation of the joint spectrum is much faster than the computation of the spectrum of ${\hat{P}}_{ε}$ from Section 5.1, since instead of a matrix of size at least $O [(E^{2} / ℏ^{2}) \times (E^{2} / ℏ^{2})]$ , we have $N = O (E / ℏ)$ matrices of sizes $O [(E / ℏ) \times (E / ℏ)]$ , so we gain (at least) an order of magnitude in $\frac{E}{/} ℏ$ . For instance, with $M = 41$ , instead of a $903 \times 903$ matrix, we have 10 matrices of sizes $1 \times 1, 2 \times 2, 3 \times 3, \dots, 6 \times 6$ .

5.4. The Case $γ = 0$

The case $γ = 0$ is interesting because the remaining terms of order 3 in the potential $V_{ε}$ , namely $x_{1}^{3}$ and $x_{1} x_{2}^{2}$ , are completely canceled out by the Birkhoff normal form: we have $μ = 0$ in Theorem 3.6. Therefore, the spectrum of ${\hat{P}}_{ε}$ is approximated up to $O_{4}$ merely by the quadratic term ${\hat{H}}_{2, 0} + ε {\hat{L}}_{2}$ , whose spectrum is explicit:

{ℏ (ω_{1, 0} (α_{1} + 2 α_{2} + \frac{3}{2})) + ℏ ε (ω_{1, 1} (α_{1} + \frac{1}{2}) + ω_{2, 1} (α_{2} + \frac{1}{2})), (α_{1}, α_{2}) \in N^{2}} .

When

ε = 0

, the theoretical results of Charles and Vũ Ng

ọ

c (2008) apply. In particular, in the regime

E = C ℏ

, the spectrum should converge to the polyads of (24), with an error of order

O (ℏ^{2})

. This is clearly illustrated in Figures 3 and 4.

Figure 3.

Comparison of the spectrum of ${\hat{P}}_{ε}$ with the eigenvalues obtained from the Birkhoff normal form ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ . Here $ℏ = 0.005$ (left) or $ℏ = 0.0005$ (right), and $γ = 0, ε = 0$ , hence ${\hat{K}}_{3, ε} = 0$ .

Figure 4.

Error(log scale) between the spectrum of ${\hat{P}}_{ε}$ and that of ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ . Here $γ = 0, ε = 0$ . When $ℏ$ is small enough, we observe the theoretical slope of 2 (green line), corresponding to an error of order $O (ℏ^{2})$ .

In order to experiment the case $ε \neq 0$ , we chose the regime $ε = \sqrt{ℏ}$ , where we still expect an error of order $O (ℏ^{2})$ , which is confirmed by the numerics, see Figures 5 and 6.

Figure 5.

Figure 6.

Error(log scale) between the spectrum of ${\hat{P}}_{ε}$ and that of ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ . Here $γ = 0, ε = \sqrt{h}$ . When $ℏ$ is small enough, we tend to the theoretical slope of 2 (green line), corresponding to an error of order $O (ℏ^{2})$ , though it seems slower than in the case $ε = 0$ (Figure 4); on the other hand, for the larger $ℏ$ , the absolute error is quite smaller here than for the case $ε = 0$ .

Figure 7.

Joint spectrum of $(ℏ^{- 1} {\hat{H}}_{2, 0}, {\hat{K}}_{3, ε})$ . The abscissae are $n + (3 / 2)$ , $n \geq 0$ . Here $γ = 1, c = 1$ , $ε = 0.01, ℏ = 0.001$ .

The joint spectrum of the commuting operators $(ℏ^{- 1} {\hat{H}}_{2, 0}, {\hat{K}}_{3, ε})$ is depicted in Figure 7.

5.5. The Case

γ = 1

The case $γ = 1$ corresponds to the heart of our result, since the Birkhoff term of order 3, ${\hat{K}}_{3, ε}$ is not trivial, due to (22).

When $ε = 0$ , as above, the theoretical results of Charles & Vũ Ngọc (2008) apply and we observe the expected $O (ℏ^{2})$ error in Figures 8 and 9. The clear agreement with $O (ℏ^{2})$ is a strong confirmation of the validity of the Birkhoff procedure, and in particular of the correctness of the value of $μ$ from (22), because any other value of $μ$ would lead to an error of the order of the eigenvalues of ${\hat{K}}_{3, 0}$ on the given spectral subspace, which is known to be $O (ℏ^{3 / 2})$ .

Figure 8.

Figure 9.

Error(log scale) between the spectrum of ${\hat{P}}_{ε}$ and that of ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ . Here $γ = 1, ε = 0$ . We observe a nice fit with the theoretical slope of 2 (green line), corresponding to an error of order $O (ℏ^{2})$ .

The new results correspond to $ε \neq 0$ . As in the case $γ = 0$ we experiment the regime $ε = \sqrt{ℏ}$ , and, in spite of this perturbation, the $ε$ -Birkhoff–Gustavson procedure suggests that the error should still be $O (ℏ^{2})$ . This is confirmed by Figures 10 and 11.

Figure 10.

Figure 11.

Error(log scale) between the spectrum of ${\hat{P}}_{ε}$ and that of ${\hat{H}}_{2, 0} + {\hat{K}}_{3, ε}$ . Here $γ = 1, ε = \sqrt{ℏ}$ . When $ℏ$ is small enough, we tend to the theoretical slope of 2 (green line), corresponding to an error of order $O (ℏ^{2})$ , though it seems slower than in the case $ε = 0$ (Figure 9); on the other hand, for the larger $ℏ$ , the absolute error is quite smaller here than for the case $ε = 0$ .

Footnotes

Acknowledgements

Kaoutar Ghomari acknowledges the excellent working conditions that she was given during several stays at the IRMAR.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported in part by funds provided by Henri Lebesgue Center (ANR LEBESGUE), the Laboratory of Fundamental and Applied Mathematics of Oran and the Algerian research project: PRFU No. C00L03ES310120220001.

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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Semiclassical Birkhoff–Gustavson Normal Forms and Spectral Asymptotics for Nearly Resonant Schrödinger Operators

Abstract

Keywords

1. Introduction and Motivation

3.1. ε -Birkhoff–Gustavson Normal Form Theorem

4.1. Creation and Annihilation Operators

Theorem 4.1 Bargmann, 1961

Proposition 4.1 Bargmann, 1961

4.3. Spectrum in Near Fermi Resonance

5.4. The Case γ = 0

Footnotes

Acknowledgements

Funding

Declaration of Conflicting Interests

References

3.1. $ε$ -Birkhoff–Gustavson Normal Form Theorem

5.4. The Case $γ = 0$