Inverse resonance scattering on rotationally symmetric manifolds

Abstract

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0, \infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove

Asymptotics of counting function of resonances at large radius.

The rotation radius is uniquely determined by its eigenvalues and resonances.

There exists an algorithm to recover the rotation radius from its eigenvalues and resonances.

The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.

Keywords

Inverse resonance scattering rotationally symmetric manifolds iso-resonance sets

1. Introduction and main results

1.1. Geometry

We consider a rotationally symmetric manifold $M = R_{+} \times Y$ equipped with a warped product metric $\begin{matrix} (1.1) & g = {(d x)}^{2} + r^{2} (x) g_{Y}, x ⩾ 0 . \end{matrix}$ Here $(Y, g_{Y})$ is a compact m-dimensional Riemannian manifold (with or without boundary), called transversal manifold, and $r (x) > 0$ is the rotation radius satisfying $\begin{matrix} (1.2) & r (x) = r_{o} = const > 0 for x ⩾ 1 . \end{matrix}$ We introduce $q (x)$ by $\begin{matrix} (1.3) & \begin{matrix} r = r_{o} e^{\frac{2}{m} Q}, Q (x) = - \int_{x}^{1} q (t) d t, x \in R_{+} = [0, \infty), \end{matrix} \end{matrix}$ and assume that the real function q belongs to the class $P_{1}$ , where $P_{j}$ is defined by $\begin{matrix} (1.4) & P_{j} = {g; g^{(j)} \in L_{real}^{1} (R_{+}), supp g \subset [0, 1], sup supp g = 1}, j ⩾ 0 . \end{matrix}$ As will be seen below, the geometry (i.e., the rotation radius r and hence all derived quantities up to two integration constants) is determined by q. The Laplacian $Δ_{M}$ on M has the form $\begin{matrix} - Δ_{M} = - \frac{1}{r^{m}} \partial_{x} (r^{m} \partial_{x}) - \frac{Δ_{Y}}{r^{2}}, \end{matrix}$ where $Δ_{Y}$ is the Laplacian on Y. We assume the Dirichlet boundary condition, i.e. the domain of $Δ_{M}$ consists of the functions $f = f (x, y)$ , $(x, y) \in M = R_{+} \times Y$ , satisfying the following boundary condition $\begin{matrix} (1.5) & \begin{matrix} f |_{\partial M} = 0, \partial M = {0} \end{matrix} \end{matrix}$ The negative Laplacian $- Δ_{Y}$ on Y (with suitable boundary conditions when Y has a boundary) has a discrete spectrum, $0 ⩽ E_{1} ⩽ E_{2} ⩽ \dots$ , with corresponding orthonormal family of eigenfunctions $Ψ_{ν}, ν ⩾ 1$ , in $L^{2} (Y)$ .

Let us recall simple examples.

The case when Y has no boundary. For example, if $Y = S^{1}$ , then $E_{1} = 0$ , $E_{2} = π^{2}, \dots$ .

The case when Y has a boundary. For example, if Y is a compact interval in $R^{1}$ , e.g. $[0, 1]$ , then $E_{1} = 0, E_{2} = π^{2}, \dots$ , for the Neumann boundary condition, and $E_{1} = π^{2}, E_{2} = {(2 π)}^{2}, \dots$ , for the Dirichlet boundary condition.

Introduce the Hilbert spaces $L_{ν}^{2} (M)$ , $ν ⩾ 1$ by $\begin{matrix} L_{ν}^{2} (M) = {h (x, y) = f (x) Ψ_{ν} (y) : (x, y) \in M = R_{+} \times Y, \int_{0}^{\infty} {| f (x) |}^{2} d x < \infty} . \end{matrix}$ The Laplacian on $(M, g)$ on $L^{2} (M) = ⨁_{ν ⩾ 1} L_{ν}^{2} (M)$ , due to the spectral decomposition of $- Δ_{Y}$ , is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators $H_{ν}$ , namely, $\begin{matrix} (1.6) & - Δ_{M} ≃ ⨁_{ν = 1}^{\infty} (H_{ν} + u_{ν, 0}) . \end{matrix}$ Here the operator $H_{ν}$ acts on $L^{2} (R_{+})$ under the Dirichlet boundary condition and is given by $\begin{matrix} (1.7) & \begin{matrix} H_{ν} f = - f^{″} + p_{ν} f, f (0) = 0, \end{matrix} \end{matrix}$ where the potential $p_{ν} (x)$ is derived in Section 2.2 below and has the form $\begin{matrix} (1.8) & \{\begin{matrix} p_{ν} (x) = q^{'} (x) + q^{2} (x) + u_{ν} (x) - u_{ν, 0}, \\ u_{ν} (x) = u_{ν, 0} e^{- \frac{4}{m} Q (x)}, Q (x) = - \int_{x}^{1} q (t) d t, \\ u_{ν, 0} = \frac{E_{ν}}{r^{2} (1)} . \end{matrix} \end{matrix}$ Note that $\begin{matrix} supp p_{ν} \subset [0, 1], p_{ν} \in L^{1} (0, 1) . \end{matrix}$

Below we fix an index ν arbitrarily, and omit it. We consider the operator $H f = - f^{″} + p f$ , where the potential $p = p_{ν}$ is given by (1.8). It is well-known (see Theorem 2.1 in [15]) that H has purely absolutely continuous spectrum $[0, \infty)$ and a finite number $N ⩾ 0$ of bound states $E_{1} < \dots < E_{N} < 0$ below the continuum. The Jost solution $f_{+} (x, k)$ is the solution to the equation $\begin{matrix} (1.9) & - f_{+}^{″} + p f_{+} = k^{2} f_{+}, x ⩾ 0, k \in C ∖ {0} \end{matrix}$ satisfying the condition $f_{+} (x, k) = e^{i x k}, x ⩾ 1$ . The Jost function $ψ (k)$ is defined by $ψ (k) = f_{+} (0, k)$ . Let $n_{+} (f)$ be the number of zeros of f analytic in the upper half pane $C_{+} = {z \in C : Im z > 0}$ , each zero being counted according to its multiplicity. For a function f analytic in a neighborhood of 0, we define $n_{o} (f) = m$ , if $f (z) = z^{m} g (z)$ where $g (0) \neq 0$ . The Jost function $ψ (k)$ has the following standard properties:

The Jost function $ψ (k)$ is entire on $C$ and has the following asymptotics uniformly in $arg k \in [0, π]$ (see e.g. (3.1.20) from [43]): $\begin{matrix} (1.10) & ψ (k) = 1 + O (1 / k) as | k | \to \infty . \end{matrix}$ uniformly in $arg k \in [0, π]$ . Due to (1.10) we define the branch of $arg ψ (k), k \in {\overline{C}}_{+} ∖ [0, k_{1}]$ by the condition $arg ψ (k) = o (1)$ as $| k | \to \infty$ .

The S-matrix is defined by $\begin{matrix} (1.11) & S (k) = \frac{ψ (- k)}{ψ (k)} = e^{- i 2 ϕ_{sc} (k)}, ϕ_{sc} (k) : = arg ψ (k), k \in R ∖ {0} . \end{matrix}$ We call $ϕ_{sc}$ the phase shift. Note that $S (k)$ is meromorphic on $C$ . We define a resonance of H to be a pole of $S (k)$ in ${\overline{C}}_{-}$ , where $C_{-} = {z \in C : Im z < 0}$ is the lower half pane.

The asymptotics (1.10) and the fact that ψ is entire imply that the number of eigenvalues is finite, since the zeros of ψ in $C_{+}$ are given by $\begin{matrix} (1.12) & k_{1} = i | E_{1} |^{\frac{1}{2}}, \dots, k_{n_{+}} = i | E_{N} |^{\frac{1}{2}} \in i R_{+}, N = n_{+} (ψ) . \end{matrix}$ There may be a simple zero of ψ at 0, hence $n_{o} (ψ) = 0$ or 1. The number of zeros in $C_{-} \cup {0}$ are infinite, and arranged so that $0 ⩽ | k_{N + 1} | ⩽ | k_{N + 2} | ⩽ \dots$ (counted with multiplicities). If $k_{N + 1} = 0$ , then $0 < | k_{N + 2} |$ . The zeros of ψ in $C_{-}$ coincide with the resonances.

For each $k_{j} \in C_{+}$ , $j = 1, \dots, N$ we define the norming constants $c_{j} > 0$ by $\begin{matrix} (1.13) & c_{j} = \int_{R_{+}} {| f_{+} (x, k_{j}) |}^{2} d x < \infty, k_{j} \in C_{+} . \end{matrix}$ Marchenko [42] introduced the norming constants $c_{j}$ to solve inverse scattering problem. Note that $f_{+} (\cdot, k_{j}) \in L^{2} (R_{+})$ for each $k_{j} \in C_{+}$ is an eigenfunction of H, since $f_{+} (0, k_{j}) = 0$ .

Since $ψ (k)$ is real on $i R_{+}$ and has $n_{+}$ zeros in $C_{+}$ , the function $ϕ_{sc}$ is odd, i.e. $ϕ_{sc} (- k) = - ϕ_{sc} (k)$ for $k > 0$ , continuous on $(0, \infty)$ , $ϕ_{sc} (k) = o (1)$ as $k \to \infty$ and $\begin{matrix} (1.14) & ϕ_{sc} (\pm 0) = \mp π (N + \frac{n_{o}}{2}), n_{o} = n_{o} (ψ) = 0 or 1 . \end{matrix}$ The fact that ψ has a zero of order at most one at zero follows from (3.20) in Levinson’s paper or from Lemma 3.1.6 from [43]. Note that this follows from the Wronskian ${f_{+} (x, - k), f_{+} (x, k)} = 2 i k$ for all $k \in C$ .

We shortly discuss well known facts about 1dim inverse scattering for Schrödinger operators with decaying potentials. Bargmann [3,4] constructed explicit examples, where two distinct potentials give the same S-matrix and the same eigenvalue (only one). This showed that a potential cannot be reconstructed uniquely from very natural spectral data: S-matrix $S (k), k > 0$ and eigenvalues. Levinson [40] proved that the reconstructed potential is unique when there is no eigenvalues. This problem was solved by Marchenko [42]. He defined a new spectral data (the phase shift $ϕ_{sc}$ plus eigenvalues plus norming constants): $\begin{matrix} (1.15) & S_{SD} = {ϕ_{sc} (k), k > 0, k_{j} \in i R_{+}, c_{j} > 0, j = 1, \dots, N} . \end{matrix}$ Marchenko proved the nice result: the mapping from potentials to their spectral data is a bijection (for specific classes of potentials and spectral data) and the algorithm to reconstruct the potential is given. Similar results were obtained by Krein [38]. There are more information about it in the books: Marchenko [43] and Chadan and Sabatier [7].

We describe a preliminary result about eigenvalues of the Laplacian $Δ_{M}$ .

Theorem 1.1.

Let $q \in P_{1}$ and p be given by ( 1.8 ). Then $p \in P_{0}$ .

If $r (x) ⩽ r_{o} = r (1)$ for all $x \in [0, 1]$ , then the Laplacian $- Δ_{M}$ has no eigenvalues.

There exist a radius function $r (x)$ and a number and $ν_{o} > 0$ such that

$r (x) > r_{o}$ for all $x \in (0, 1)$ ,

each operator $Δ_{ν}$ , $ν > ν_{o}$ , has eigenvalues,

the Laplacian $Δ_{M}$ has infinite number of eigenvalues.

Remark.

It is important for inverse resonance scattering to show that $p \in P_{0}$ .

The asymptotic behavior of the distribution function of the eigenvalues of the Laplacian on warped product manifolds with cylindrical ends are discussed in [10,45].

Fig. 1.
(a) Δ has eigenvalues, (b) Δ has not eigenvalues.

We consider the distribution of resonances of $- Δ_{M}$ . For an analytic function f on $C$ , let $N_{r} (f)$ be the number of zeros of f counted with multiplicities in the disc of radius r centered at the origin. By virtue of Theorem 1.1, (i) and the well-known result of Zworski [49] (see also [28,32,47]) for the distribution of resonances, we obtain.
Corollary 1.2.
Let $q \in P_{1}$ . Then $\begin{matrix} (1.16) & N_{r} (ψ) = \frac{2 r}{π} (1 + o (1)) as r \to \infty . \end{matrix}$

Let $n_{+} (f)$ be the number of zeros of f in $C_{+}$ counted with multiplicities. Define the resolvents $R_{o} (k) = {(H_{o} - k^{2})}^{- 1}$ and $R (k) = {(H - k^{2})}^{- 1}$ , $k \in C_{+}$ .
Corollary 1.3.
Let $q \in P_{1}$ . Then the Jost function $ψ (k)$ is represented as $\begin{matrix} (1.17) & ψ (k) = k^{s} ψ^{(s)} (0) e^{i k} lim_{r \to \infty} \prod_{| k_{n} | ⩽ r, k_{n} \neq 0} (1 - \frac{k}{k_{n}}), k \in C, \end{matrix}$ uniformly on any compact subset of $C$ , where $ψ^{(s)} (0) = \frac{\partial^{s}}{\partial k^{s}} ψ (0)$ and $s = n_{o} (ψ) \in {0, 1}$ . Moreover, the S-matrix has the form $S (k) = e^{- 2 i ϕ_{sc} (k)}$ , where $ϕ_{sc}$ is given by $\begin{matrix} (1.18) & \begin{matrix} ϕ_{sc} (k) = - π (n_{+} + \frac{n_{o}}{2}) + \int_{0}^{k} ϕ_{sc}^{'} (t) d t, \\ ϕ_{sc}^{'} (k) = 1 + \sum_{n = 1}^{\infty} \frac{Im k_{n}}{| k - k_{n} |^{2}}, k > 0, \end{matrix} \end{matrix}$ uniformly on any compact subset of $R_{+}$ , where $n_{+} = n_{+} (ψ)$ . Furthermore, the norming constants $c_{j} > 0$ are given by $\begin{matrix} (1.19) & c_{j} = - i \frac{ψ^{'} (k_{j})}{ψ (- k_{j})}, j = 1, \dots, N, \end{matrix}$ and the following trace formula holds true: $\begin{matrix} (1.20) & - 2 k Tr (R (k) - R_{o} (k)) = \frac{n_{o}}{k} + i + lim_{r \to \infty} \sum_{k_{n} \neq 0, | k_{n} | < r} \frac{1}{k - k_{n}}, \end{matrix}$ uniformly on compact subsets of $C ∖ {0, k_{n}, n \in N}$ .
Remark.

Using zeros of the Jost function ψ we determine the phase shift $ϕ_{sc}$ , plus eigenvalues $k_{j}$ , $j = 1, \dots, N$ plus norming constants $c_{j}$ , $j = 1, \dots, N$ . Thus we have all spectral data from (1.15) to determine a potential p via Marchenko results [43], see also Section 3 about it. The function q we will determine using

The constant $ψ^{(s)} (0)$ from (1.17) is absent in the identities (1.18) and (1.19). Thus we do not need one to compute $ϕ_{sc}$ and norming constants $c_{j}$ , $j = 1, \dots, N$ .

1.2. Inverse problem

We sometimes write $ψ (k, q), k_{n} (q), \dots$ instead of $ψ (k), k_{n}, \dots$ , when several potentials are dealt with. We introduce the Sobolev space of real functions $\begin{matrix} (1.21) & \begin{matrix} W_{1}^{0} = {q \in L_{real}^{2} (0, 1); q^{'} \in L^{2} (0, 1), q (0) = q (1) = 0}, \end{matrix} \end{matrix}$ equipped with norm $‖ q ‖_{W_{1}^{0}}^{2} = ‖ q^{'} ‖^{2} = \int_{0}^{1} | q^{'} (x) |^{2} d x$ .

The main result of this paper is the following inverse problem.

Theorem 1.4.

Let $ψ_{j}$ be the Jost function for $q_{j} \in W_{1}^{0} \cap P_{1}$ , $j = 1, 2$ . If $ψ_{1} = ψ_{2}$ , then $q_{1} = q_{2}$ .

Let $S_{j}$ be the S-matrix for $q_{j} \in W_{1}^{0} \cap P_{1}$ , $j = 1, 2$ . If $S_{1} = S_{2}$ , then $q_{1} = q_{2}$ .

Any $q \in W_{1}^{0} \cap P_{1}$ is uniquely determined by its eigenvalues and resonances. Moreover, there exists an algorithm to recover q from its eigenvalues and resonances.

Remark.
The proof is based on the two mappings, since the mapping q to a Jost function ψ is the composition of two mappings. The first one is the mapping from potentials $p_{ν}$ to the Jost functions (or the S-matrices), see [28] and Theorem 3.2. The second one is the mapping $q \to p_{ν}$ from (1.8) given by $p_{ν} (x) = q^{'} (x) + q^{2} (x) + u_{ν} (x) - u_{ν, 0}$ .

1.3. Historical review

There is an abundance of works devoted to the spectral theory and inverse problems for the surface of revolution from the view points of classical inverse Strum–Liouville theory, integrable systems, micro-local analysis, see [1,5,14,19,41] and references therein. For integrable systems associated with surfaces of revolution, see e.g. [24,46,48] and references therein.

Isozaki–Korotyaev [21,22] solved the inverse spectral problem for rotationally symmetric manifolds (finite perturbed cylinders), which includes a class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the radius of rotation. An analogue of the Minkowski problem was also solved. In another paper [23] Isozaki–Korotyaev studied inverse problems for Laplacian on the torus. Moreover, they obtained stability estimates: the spectral data in terms of the profile (the radius of the rotation) and conversely, the profile in term of the spectral data.

In our paper we use inverse resonance theory for Schrödinger operators with compactly supported potentials on the half-line [28]. We use also results on perturbed Riccati mappings from [22,26,27].

A lot of papers are devoted to resonances of one-dimensional Schrödinger operators. See Froese [17], Hitrik [20], Korotyaev [28], Simon [47], Zworski [49] and references therein. Inverse problems (uniqueness, reconstruction, characterization) in terms of resonances were solved by Korotyaev for a Schrödinger operator with a compactly supported potential on the real line [30] and the half-line [28]. See also Zworski [50], Brown-Knowles-Weikard [6] concerning the uniqueness. The resonances for one-dimensional operators $- \frac{d^{2}}{d x^{2}} + V_{π} + V$ , where $V_{π}$ is periodic and V is a compactly supported potential were considered by Firsova [16], Korotyaev [31], Korotyaev–Schmidt [37]. Christiansen [9] considered resonances for steplike potentials. The “local resonance” stability problems were considered in [29,32,44]. Resonances for three and fourth order differential operator with compactly supported coefficients on the line were studied in [2,34]. Resonances for Stark operator with compactly supported potentials were discussed in [18,33]. Inverse problems terms of resonances (uniqueness, reconstruction, characterization) for Dirac operators with a compactly supported potential on the half line was discussed in [36].

Resonances for specific cases of surfaces of revolution are discussed in [8,11–13]. As far as the authors know, the results in this paper about resonances for Laplacian on surfaces of revolution for the case (1.1)–(1.4) are new.

2. Preliminaries

2.1. Entire functions

We recall some well-known facts about entire functions (see [25]). An entire function $f (k)$ is said to be of exponential type if there is a constant α such that $| f (k) | ⩽$ const. $e^{α | k |}$ everywhere on $C$ . An entire function f is said to belong to the Cartwright class $E_{Cart}$ , if $f (k)$ is of exponential type, and the following conditions hold true: $\begin{matrix} (2.1) & \begin{matrix} \int_{R} \frac{{log}^{+} | f (x) | d x}{1 + x^{2}} < \infty, \\ ϱ_{+} (f) = 0, ϱ_{-} (f) = 2, where ϱ_{\pm} (f) = lim sup_{t \to \infty} \frac{log | f (\pm i t) |}{t} . \end{matrix} \end{matrix}$ Denote by ${(k_{n})}_{n = 1}^{\infty}$ the sequence of zeros $\neq 0$ of f (counted with multiplicities), arranged so that $0 < | k_{1} | ⩽ | k_{2} | ⩽ \dots$ . We recall the main properties of functions $f \in E_{Cart}$ (see theorem Cartwright and Levinson see p. 127 in [39]):

there exists the limit $\begin{matrix} (2.2) & lim_{r \to + \infty} \sum_{| k_{n} | ⩽ r} \frac{1}{k_{n}} . \end{matrix}$

the following sum is finite $\begin{matrix} (2.3) & \sum_{1}^{\infty} \frac{| Im k_{n} |}{| k_{n} |^{2}} < \infty . \end{matrix}$

Let $N_{r} (f)$ be the number of zeros of $f (k)$ in the disk $| k | ⩽ r$ (counting multiplicities). Then the following asymptotics of Levinson (see [25]) holds true: $\begin{matrix} (2.4) & N_{r} (f) = \frac{2}{π} r + o (r) as r \to \infty . \end{matrix}$

a function $f \in E_{Cart}$ has the Hadamard factorization $\begin{matrix} (2.5) & f (k) = k^{s} C e^{i k} lim_{r \to + \infty} \prod_{| k_{n} | ⩽ r} (1 - \frac{k}{k_{n}}), C = \frac{f^{(s)} (0)}{s!}, \end{matrix}$ for some integer $s ⩾ 0$ , where the product converges uniformly on any compact set. We remark that if f is exponential type, then there is the additional factor $e^{k / k_{n}}$ in (2.5). But in the case $f \in E_{Cart}$ we can remove this factor in (2.5), since there exists the limit (2.2).

From (2.5) we obtain the identity $\begin{matrix} (2.6) & \frac{f^{'} (k)}{f (k)} = i + \frac{s}{k} + lim_{R \to \infty} \sum_{| k_{n} | ⩽ R} \frac{1}{k - k_{n}} \end{matrix}$ uniformly on compact subsets of $C ∖ {0, k_{n}, n = 1, 2, \dots}$ .

In our paper, we make use of following functions from $E_{Cart}$ : if $g \in P_{0}$ , then we have $\begin{matrix} (2.7) & 1 + \hat{g} \in E_{Cart}, where \hat{g} (k) = \int_{0}^{1} g (t) e^{2 i k t} . \end{matrix}$

2.2. Unitary transformations

Recall that the Laplacian on $(M, g)$ acting in $L^{2} (M) = ⨁_{ν ⩾ 1} L_{ν}^{2} (M)$ is unitarily equivalent to a direct sum (1.6) of one-dimensional operators $Δ_{ν}$ . Here the operator $Δ_{ν}$ acts in the space $L^{2} (R_{+}, r^{m} (x) d x)$ under the Dirichlet boundary condition and is given by $\begin{matrix} (2.8) & \begin{matrix} - Δ_{ν} f = - \frac{1}{r^{m}} {(r^{m} f^{'})}^{'} + u_{ν} f, f (0) = 0, \\ u_{ν} = \frac{E_{ν}}{r^{2}} = u_{ν, 0} e^{- \frac{4}{m} Q}, u_{ν, 0} = u_{ν} (1), r = r_{o} e^{\frac{2}{m} Q}, Q = - \int_{x}^{1} q (t) d t . \end{matrix} \end{matrix}$ The function $ϱ = r^{\frac{m}{2}} > 0$ is called an impedance function. We define a unitary transformation $U$ by $\begin{matrix} U : L^{2} (R_{+}, ϱ^{2} d x) \to L^{2} (R_{+}, d x), U f = ϱ f, ϱ = r^{\frac{m}{2}} = ϱ_{0} e^{Q} . \end{matrix}$ We transform the operator $- Δ_{ν}$ into a Schrödinger operator $H_{ν}$ by $\begin{matrix} (2.9) & \begin{matrix} U (- Δ_{ν}) U^{- 1} = - ϱ^{- 1} \partial_{x} ϱ^{2} \partial_{x} ϱ^{- 1} + u_{ν} = D^{*} D + u = H_{ν} + u_{ν, 0}, \end{matrix} \end{matrix}$ where $H_{ν}$ acts on $L^{2} (R_{+})$ and is given by $\begin{matrix} (2.10) & \begin{matrix} H_{ν} = - \frac{d^{2}}{d x^{2}} + p_{ν}, p_{ν} = q^{'} + q^{2} + u_{ν} - u_{ν, 0}, \end{matrix} \end{matrix}$ and where the potential $p_{ν} \in L^{1} (R_{+})$ has a compact support $\subset [0, 1]$ . Here we have used the identities $\begin{matrix} (2.11) & \begin{matrix} D = ϱ \partial_{x} ϱ^{- 1} = \partial_{x} - q, D^{*} = {(ϱ \partial_{x} ϱ^{- 1})}^{*} = - \partial_{x} - q, \\ D^{*} D = - (\partial_{x} + q) (\partial_{x} - q) = - \partial_{x}^{2} + q^{'} + q^{2} . \end{matrix} \end{matrix}$

The following support property of the non-linear mappings $q \to p_{ν}$ in (2.10) plays a key role.

Lemma 2.1.
Let $q, q^{'} \in L^{1} (Ω)$ for $Ω = (1 - τ, 1)$ for some $τ > 0$ and $q (1) = 0$ . Assume that q satisfies on Ω the following equation $\begin{matrix} (2.12) & q^{'} + q^{2} + u - u_{0} = 0, \end{matrix}$ where $u = u_{0} e^{β Q}$ with $β, u_{0} > 0$ . Then $q = 0$ on $ω = (1 - ε^{2}, 1)$ for $ε > 0$ small enough.
Proof.
Define the norm in $L^{r} (ω)$ , $r ⩾ 1$ by $‖ q ‖_{r}^{r} = \int_{ω} | q (t) |^{r} d t$ . We have for $x \in ω$ $\begin{matrix} (2.13) & q (x) = - \int_{x}^{1} q^{'} (t) d t, | q (x) | ⩽ ‖ q^{'} ‖ ε, ‖ q ‖ ⩽ ε {‖ q^{'} ‖}_{1}, \end{matrix}$ where $‖ q ‖ = ‖ q ‖_{2}$ . Let $q \neq 0$ . Then for $ε > 0$ small enough this yields $\begin{matrix} (2.14) & \begin{matrix} ‖ q ‖ ⩽ 1, | Q (x) | ⩽ \int_{x}^{1} | q (t) | d t ⩽ ε ‖ q ‖ ⩽ ε, \\ | e^{β Q (x)} - 1 | = β | \int_{x}^{1} q (t) e^{β Q (t)} d t | ⩽ β e^{β ε} \int_{x}^{1} | q (t) | d t ⩽ ε β ‖ q ‖ e^{β ε}, \end{matrix} \end{matrix}$ for all $x \in ω$ . Using these estimates for (2.12), we have $\begin{matrix} (2.15) & \begin{matrix} q (x) = - \int_{x}^{1} (q^{2} + u - u_{0}) d t, \\ | q (x) | ⩽ \int_{x}^{1} ({| q (t) |}^{2} + u_{0} | e^{β Q (x)} - 1 |) d t ⩽ ‖ q ‖^{2} + ε β ‖ q ‖ u_{0} e^{β ε} ⩽ ‖ q ‖ C, \end{matrix} \end{matrix}$ where $C = 1 + ε β u_{0} e^{β ε}$ . This gives $‖ q ‖ ⩽ ε ‖ q ‖ C$ , which yields a contradiction, since $ε C < 1$ for $ε > 0$ small enough. □

3. Inverse resonance scattering on the half-line

3.1. Resonance scattering

We consider the Schrödinger operator on $L^{2} (R_{+})$ given by $\begin{matrix} (3.1) & H y = - y^{″} + p (x) y, y (0) = 0, \end{matrix}$ where the potential $p \in P_{0}$ . It is well-known (see Theorem 1.2 in [15]) that H has purely absolutely continuous spectrum $[0, \infty)$ plus a finite number $n_{+} ⩾ 0$ of bound states $E_{1} < \dots < E_{n_{+}} < 0$ . Introduce the Jost solution $f_{+} (x, k)$ and the regular solution $φ (x, k)$ of the equation $\begin{matrix} (3.2) & - y^{″} + p y = k^{2} y, x ⩾ 0, k \in C ∖ {0}, \end{matrix}$ with the conditions $f_{+} (x, k) = e^{i x k}, x ⩾ 1$ and $φ (0, k) = 0$ , $φ^{'} (0, k) = 1$ . Outside the support of p they are linear combinations of $e^{\pm i k x}$ . We define the Jost function ψ by $\begin{matrix} (3.3) & ψ (k) = f_{+} (0, k) = 1 + \int_{0}^{1} \frac{sin k x}{k} p (x) f_{+} (x, k) d x, k \in C . \end{matrix}$ This $ψ (k)$ has the well known properties (see e.g., [28]), which we summarize here.

Lemma 3.1.
If $p \in P_{0}$ , the Jost function $ψ (k)$ for the operator $- y^{″} + p y$ on the half-line with condition $y (0) = 0$ has the form $\begin{matrix} (3.4) & ψ (k) = 1 + \frac{\hat{F} (k) - \hat{F} (0)}{2 i k}, \forall k \in C, \end{matrix}$ for some $F \in P_{0}$ . Moreover, the function ψ satisfies
$ψ (k) = 1 + \frac{O (1)}{| k |}$ as $| k | \to \infty$ , $k \in {\overline{C}}_{+}$ , uniformly in $arg k \in [0, π]$ .

$ψ (k)$ is real on $i R_{+}$ ,

$ψ (k)$ has $n_{+}$ simple zeros in $C_{+}$ , which belong to $i R_{+}$ ,

the function $ϕ_{sc}$ is odd $ϕ_{sc} (- k) = - ϕ_{sc} (k)$ , $k > 0$ , and is continuous on $(0, \infty)$

$ϕ_{sc} (k) = o (1)$ as $k \to \infty$ and $\begin{matrix} (3.5) & ϕ_{sc} (\pm 0) = \mp π (n_{+} + \frac{n_{o}}{2}), as n_{o} = n_{o} (ψ) = 0, 1 . \end{matrix}$

Let $Z (f)$ be the set of zeros of an entire function f. The zeros of ψ satisfies $\begin{matrix} (3.6) & Z (ψ) \cap Z (ψ (- \cdot)) = \{\begin{matrix} {0} & if ψ (0) = 0, \\ \emptyset & if ψ (0) \neq 0 . \end{matrix} \end{matrix}$

Proof.
The identity (3.4) was proved in [28]. Asymptotics in (i) follows from (3.4), it is the same as (1.10). The properties (2)–(5) are well know, see e.g., [43] or [15].

Note that (2) holds true, since it comes from the solution of differential equation with real coefficients.

Recall that $ψ (k) = f_{+} (0, k) = {f_{+} (\cdot, k), φ (\cdot, k)}$ . Thus if $ψ (k_{o}) = 0$ for some $k_{o} \in i R_{+}$ , then $f_{+} (\cdot, k)$ and $φ (\cdot, k)$ are linearly dependent and $k_{o}^{2}$ is an eigenvalue of H. Thus in (3), zeros $k_{j} \in C_{+}$ are given by $k_{j} = i | E_{j} |^{\frac{1}{2}}$ , $j = 1, \dots, N$ .

Item (4) follows from $ψ (- k) = \overline{ψ} (k)$ , $k \in R$ .

Recall that $ϕ_{sc} (k) = arg ψ (k)$ , where the branch of $arg ψ$ is chosen by $arg ψ (k) = o (1)$ as $k \to \infty$ . It is possible since $ψ (k) = 1 + O (1 / k)$ as $k \to \infty$ and $ψ (k) \neq 0$ and ψ is continuous on $k > 0$ . Item (5) follows from Section 4 of Levinson’s paper [40], or from (3.2.22) of [43].

The property (6) was proved in [28]. In fact it this follows from the Wronskian ${f_{+} (x, - k), f_{+} (x, k)} = 2 i k$ for all $k \in C$ . □

Introduce the set $J$ of all possible Jost functions from [28].
Definition $J$ .
By $J$ we mean the class of all entire functions f having the form $\begin{matrix} (3.7) & \begin{matrix} f (k) = 1 + \frac{1}{2 i k} (\hat{F} (k) - \hat{F} (0)), k \in C, \end{matrix} \end{matrix}$ where $\hat{F} (k) = \int_{0}^{1} F (x) e^{2 i x k} d x$ and $F \in P_{0}$ . In addition the sequence ${(k_{n})}_{n = 1}^{\infty}$ of zeros of f satisfies:
the function $f (k) \neq 0$ for any $k \in R ∖ {0}$ and $n_{o} (f) ⩽ 1$ ,

all zeros $k_{1}, \dots, k_{n_{+}}$ of the function f in $C_{+}$ are simple, belong to $i R_{+}$ and satisfy $\begin{array}{l} (3.8) & \begin{aligned} | k_{1} | > | k_{2} | > \dots > | k_{n_{+}} | > 0, \\ {(- 1)}^{j} f (- k_{j}) > 0, j = 1, \dots, n_{+} = n_{+} (f) . \end{aligned} \end{array}$

We recall the well-known Marchenko result [43] about the inverse problems for our case $p \in P_{0}$ . Define the functions $\begin{matrix} (3.9) & G_{0} (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} (1 - S (k)) e^{i x k} d k, G (x) = G_{0} (x) + \sum_{k_{j} \in C_{+}} \frac{e^{- x | k_{j} |}}{c_{j}}, \end{matrix}$ and the norming constants $\begin{matrix} (3.10) & c_{j} = \int_{R_{+}} {| f_{+} (x, k_{j}) |}^{2} d x < \infty, k_{j} \in C_{+}, j = 1, \dots, n_{+}, \end{matrix}$ where $f_{+} (x, k_{j}) \in L^{2} (R_{+})$ , $k_{j} \in C_{+}$ is an eigenfunction of H and $f_{+} (x, k_{j}) = e^{- | k_{j} | x}$ for $x > 1$ . For each $x > 0$ the Marchenko equation (see Theorem 3.2.1 and Theorem 3.3.1 in [43]) $\begin{matrix} (3.11) & K (x, t) = - G (x + t) - \int_{x}^{\infty} G (t + s) K (x, s) d s, t ⩾ x, \end{matrix}$ has the unique solution $K (x, t)$ , which satisfies (3.11). Moreover, the potential p has the form $\begin{matrix} (3.12) & p (x) = - 2 \frac{d}{d x} K (x, x), x > 0, \end{matrix}$ see (3.1.6)′ in [43]. Below we use the fact: if $G (x) = 0$ for $x > 2 d$ , then $p (x) = 0$ for $x > d$ .

Now we consider the mapping $p \to J (p) = ψ (\cdot, p) \in J$ , i.e., for each $p \in P_{0}$ we have the Jost function $J (p) = ψ (k, p)$ . We are now in a position to formulating the main result on the inverse resonance scattering from [28] including characterization.
Theorem 3.2.

The mapping $J : P_{0} \to J$ given by $p \to J (p) = ψ (\cdot, p)$ is one-to-one and onto.

For any $p \in P_{0}$ the Jost function $ψ (k) = ψ (k, p)$ is given by $\begin{matrix} (3.13) & ψ (k) = k^{s} ψ^{(s)} (0) e^{i k} lim_{r \to \infty} \prod_{| k_{n} | ⩽ r, k_{n} \neq 0} (1 - \frac{k}{k_{n}}), k \in C, s = n_{o} (ψ) = {0, 1}, \end{matrix}$
where $ψ^{(s)} (0) = \frac{\partial^{s}}{\partial k^{s}} ψ (0)$ and the product is uniformly convergent on any compact subset of $C$ . Moreover, the potential p is uniquely determined by the Marchenko equations ( 3.11 )–( 3.12 ) (in terms of its eigenvalues and resonances), where $S (k) = e^{- 2 i ϕ_{sc} (k)}$ and $ϕ_{sc}$ is given by $\begin{matrix} (3.14) & \begin{matrix} ϕ_{sc} (k) = - π (n_{+} (ψ) + \frac{s}{2}) + \int_{0}^{k} ϕ_{sc}^{'} (t) d t, \\ ϕ_{sc}^{'} (k) = 1 + \sum_{n = 1}^{\infty} \frac{Im k_{n}}{| k - k_{n} |^{2}}, k > 0, \end{matrix} \end{matrix}$ and the norming constant $c_{j}$ , defined by ( 3.10 ), has the following form $\begin{matrix} (3.15) & c_{j} = {(- 1)}^{s} \frac{e^{- 2 | k_{j} |}}{2 | k_{j} |} \prod_{n \neq j, n ⩾ 1, k_{n} \neq 0}^{\infty} (\frac{1 - \frac{k_{j}}{k_{n}}}{1 + \frac{k_{j}}{k_{n}}}), j = 1, \dots, n_{+} (ψ) . \end{matrix}$
Remark.

The series (3.14) converges absolutely (see (2.8)).

We recall the following trace formula from [35]: $\begin{matrix} (3.16) & - 2 k Tr (R (k) - R_{0} (k)) = \frac{n_{o}}{k} + i + lim_{r \to \infty} \sum_{k_{n} \neq 0, | k_{n} | < r} \frac{1}{k - k_{n}}, \end{matrix}$
uniformly on compact subsets of $C ∖ ({0} \cup ⋃ {k_{n}})$ .

We divide the inverse problem for the mapping $k_{∙} : p \to k_{∙} (p) = {(k_{n} (p))}_{1}^{\infty}$ into three parts:

Uniqueness. Do the eigenvalues and the resonances determine uniquely p?

Reconstruction. Give an algorithm for recovering p from the sequence $k_{∙} (p) = {(k_{n} (p))}_{1}^{\infty}$ only.

Characterization. Give necessary and sufficient conditions for the sequence $k_{∙} = {(k_{n})}_{1}^{\infty}$ to be the eigenvalues and the resonances for some $p \in P_{0}$ .

Theorem 3.2 gives the solution of I and II. Moreover, it gives a characterization of $p \in P_{0}$ in terms of functions from $J$ through the mapping $J : P_{0} \to J$ . The proof of Theorem 3.2 is based on the entire function theory and the Marchenko results about the inverse scattering on the half-line.
4. Proof of main theorems

4.1. Eigenvalues and resonances

Proof of Theorem 1.1.
(i) Let $q \in P_{1}$ . We show that $p = p_{ν}$ given by (1.8) belongs to $P_{0}$ . Assume that $p \notin P_{0}$ and $p = 0$ on $Ω = (1 - τ, 1)$ for some $τ > 0$ . Then by Lemma 2.1, we have $q = 0$ on $Ω = (1 - τ_{1}, 1)$ for some small $τ_{1} > 0$ , which gives the contradiction, since $q \in P_{1}$ . Thus we have $p \in P_{0}$ .

(ii) Consider the case $r (x) ⩽ r_{o}$ for all $x \in [0, 1]$ . The Schrödinger operator $H_{ν} = H_{0} + U_{ν}$ , where due to (2.11) we have $\begin{matrix} H_{0} = D^{*} D = - (\partial_{x} + q) (\partial_{x} - q) = - \partial_{x}^{2} + q^{'} + q^{2} = - \partial_{x}^{2} + X_{0} ⩾ 0, \end{matrix}$ where $D = ϱ \partial_{x} ϱ^{- 1} = \partial_{x} - q$ and $X_{0} = q^{'} + q^{2}$ , and the potential $U_{ν}$ has the form: $\begin{matrix} (4.1) & \begin{matrix} U_{ν} = u_{ν} - u_{ν, 0} = E_{ν} (\frac{1}{r^{2}} - \frac{1}{r_{o}^{2}}) = E_{ν} \frac{r_{o}^{2} - r^{2}}{r_{o}^{2} r^{2}}, supp U_{ν} \subset [0, 1] . \end{matrix} \end{matrix}$ Recall that $u = u_{ν, 0} e^{- \frac{4 Q}{m}}$ and $Q = - \int_{x}^{1} q (t) d t$ . Then the potential $U_{ν} ⩾ 0$ and the operator $H_{ν} ⩾ H_{0} ⩾ 0$ and $H_{ν}$ has no eigenvalues. Note that $H_{0} ⩾ 0$ and the operator $H_{0}$ has only absolutely continuous spectrum. Nonnegative eigenvalues can be ruled easily by observing the following. If $H ψ = E ψ$ with $E > 0$ , then ψ is not in $L^{2} (R_{+})$ because $- ψ^{″} = E ψ$ when $x ⩾ 1$ .

(iii) Consider the case $r (x) > r_{o} = 1$ for all $x \in (0, 1)$ . We take $r (x) = 1 + \frac{{(x - 1)}^{2}}{2}$ for $x \in [0, 1]$ . Then we obtain $1 ⩽ r ⩽ \frac{3}{2}$ and $r^{'} = x - 1$ , $r^{″} = 1$ , and the identity $ϱ = r^{\frac{m}{2}}$ gives $\begin{matrix} (4.2) & \begin{matrix} q = \frac{ϱ^{'}}{ϱ} = \frac{m}{2} \frac{r^{'}}{r} = \frac{m}{2} \frac{(x - 1)}{r}, \\ q^{'} = \frac{m}{2} \frac{r^{″}}{r} - \frac{m}{2} \frac{{r^{'}}^{2}}{r^{2}} = \frac{m}{2 r} - \frac{2}{m} q^{2} = \frac{m}{2} - \frac{r}{m} q^{2} - \frac{2}{m} q^{2} = \frac{m}{2} - \frac{r + 2}{m} q^{2}, \\ X_{0} = q^{'} + q^{2} = \frac{m}{2} - \frac{r + 2 - m}{m} q^{2} \end{matrix} \end{matrix}$ and $\begin{matrix} U_{ν} = E_{ν} \frac{1 - r^{2}}{r^{2}} = - E_{ν} \frac{(1 + r) {(x - 1)}^{2}}{2 r^{2}} = - 2 E_{ν} \frac{(1 + r)}{m^{2}} q^{2} . \end{matrix}$ Let $χ_{ω}$ be the characteristic function of the interval $ω = (0, \frac{1}{2})$ . We have $q^{2} ⩾ \frac{1}{4} χ_{ω}$ . Assume that $E_{ν} ⩾ m^{2}$ . This and $p = X_{0} + U_{ν}$ yield $\begin{matrix} (4.3) & \begin{matrix} p = \frac{m}{2} - \frac{r + 2 - m}{m} q^{2} - 2 E_{ν} \frac{(1 + r)}{m^{2}} q^{2} = \frac{m}{2} - v_{ν}, \\ v_{ν} = 2 E_{ν} q^{2} {\tilde{v}}_{ν}, {\tilde{v}}_{ν} = \frac{1 + r}{m^{2}} + \frac{r + 2 - m}{m 2 E_{ν}}, \frac{1}{m^{2}} ⩽ {\tilde{v}}_{ν} ⩽ \frac{6}{m^{2}} . \end{matrix} \end{matrix}$ For $E_{ν} ⩾ m^{2}$ large enough we obtain $v_{ν} = 2 E_{ν} q^{2} {\tilde{v}}_{ν} ⩾ v_{ν}^{0} : = \frac{E_{ν}}{2 m^{2}} χ_{ω}$ . Define the operators $h_{ν}$ , $h_{ν}^{0}$ on $R_{+}$ by $\begin{array}{l} h_{ν} y = - y^{″} + (\frac{m}{2} - v_{ν}) y, \\ h_{ν}^{0} y = - y^{″} + (\frac{m}{2} - v_{ν}^{0}) y, y (0) = 0 . \end{array}$ It is well-known that the operator $h_{ν}^{0}$ has a large number of negative eigenvalues for $E_{ν}$ large enough. Then the operator $H_{ν}$ has also a large number of eigenvalues for $E_{ν}$ large enough, since $H_{ν + 1} ⩽ H_{ν} ⩽ h_{ν} ⩽ h_{ν}^{0}$ . This yields (iii). □
Proof of Corollary 1.2.
Theorem 1.1, (i) gives that $p \in P_{0}$ . In this case the well-known result of Zworski [49] (see also [28,47]) for the distribution of resonances gives the asymptotics (1.16). □
Proof of Corollary 1.3.
Theorem 1.1, (i) gives that $p \in P_{0}$ . Thus all results of Corollary 1.3 follow from Theorem 3.2 and (3.16). □
4.2. Inverse problem

We introduce the Sobolev spaces of real functions $\begin{matrix} (4.4) & \begin{matrix} H_{0} = {q \in L_{real}^{2} (0, 1) : \int_{0}^{1} q (x) d x = 0} H_{α} = {q, q^{(α)} \in H_{0},}, α ⩾ 0, \end{matrix} \end{matrix}$ equipped with norm $‖ q ‖_{H_{α}}^{2} = ‖ q^{(α)} ‖^{2} = \int_{0}^{1} | q^{(α)} (x) |^{2} d x$ .

We rewrite (1.8) in the form, omitting ν for simplicity, $\begin{matrix} (4.5) & \begin{matrix} p = v + v_{0}, v = q^{'} + q^{2} + u - u_{0} - v_{0} = q^{'} + q^{2} + u - c_{0}, \\ v_{0} = \int_{0}^{1} (q^{'} + q^{2} + u - u_{0}) d x = \int_{0}^{1} (q^{2} + u - u_{0}) d x, \\ c_{0} = u_{0} + v_{0} = \int_{0}^{1} (q^{2} + u) d x . \end{matrix} \end{matrix}$ Thus we can consider the mapping $V : W_{1}^{0} \to H_{0}$ given by $\begin{matrix} (4.6) & \begin{matrix} v = V (q) = q^{'} + q^{2} + u - c_{0}, \\ u (Q) = u_{0} e^{- β Q}, Q (x) = - \int_{x}^{1} q (t) d t, c_{0} = \int_{0}^{1} (q^{2} + u) d x, β = \frac{4}{m}, \end{matrix} \end{matrix}$ where $u_{0} = \frac{E_{ν}}{r_{o}^{2}}$ . Recall Theorem 2.3 from [23] about the mapping $q \to v = V (q)$ .

Theorem 4.1.

The mapping $V : W_{1}^{0} \to H_{0}$ , given by ( 4.6 ) is a real analytic isomorphism between the Hilbert spaces $W_{1}^{0}$ and $H_{0}$ and satisfies: $\begin{matrix} (4.7) & {‖ q^{'} ‖}^{2} ⩽ ‖ v ‖^{2} ⩽ {‖ q^{'} ‖}^{2} + 2 ‖ q ‖^{3} ‖ q^{'} ‖ + C_{*} ‖ q ‖^{2} e^{2 β ‖ q ‖}, \end{matrix}$ where the constant $C_{*} = u_{0} (β + 1) (2 + β u_{0})$ .

We consider the inverse problem: to determine a coefficient $q \in W_{0}^{1}$ , if the resonances of the operator $Δ_{ν}$ are given. Due to (1.6) it is equivalent to know the resonances of the operator $H_{ν}, p_{ν} \in H_{0}$ .

Proof of Theorem 1.4.

(i) Let $ψ_{j} (\cdot)$ be the Jost function for some $q_{j} \in W_{1}^{0} \cap P$ , $j = 1, 2$ . Assume that $ψ_{1} = ψ_{2}$ . Then Theorem 3.2 gives that $p_{1} = p_{2}$ . Thus Theorem 4.1 implies that $q_{1} = q_{2}$ .

(ii) Let $S_{j}$ be the S-matrix for some $q_{j} \in W_{1}^{0} \cap P$ , $j = 1, 2$ . Assume that $S_{1} = S_{2}$ . Let $ψ_{j}$ be the associated Jost function. Let $Z (ψ)$ be the sequence of zeros of ψ counted with multiplicity. From the identity $S_{1} (k) = \frac{ψ_{1} (- k)}{ψ_{1} (k)}$ and (3.6) we have $Z (ψ_{1}) = Z (ψ_{2})$ . Then from (3.13) and the asymptotic (1.10) we obtain $ψ_{1}^{s} (0) = ψ_{2}^{s} (0)$ and hence $ψ_{1} = ψ_{2}$ . Then, (i) gives $p_{1} = p_{2}$ . Thus Theorem 4.1 implies $q_{1} = q_{2}$ .

(iii) Let $Z (ψ)$ be the set of all eigenvalues and resonances for the Jost function $ψ (k, q)$ for some $q \in W_{1}^{0}$ and the corresponding potential p. Then we can recover the Jost function ψ by the Hadamard factorization (1.17). Note that the coefficient $ψ^{s} (0)$ is determined by asymptotics (1.10). Thus using the Jost function ψ and the Marchenko equation (see (3.9)–(3.12)), we can recover the potential p. If we know the potential p, then we can recover the unique function $q \in W_{1}^{0} \cap P$ by the perturbed Riccati equation using Theorem 4.1. □

Footnotes

Acknowledgements

E. Korotyaev was supported by the RSF grant No. 18-11-00032. H.Isozaki is partially supported by Grants-in-Aid for Scientific Research (S) 15H05740, and Grants-in-Aid for Scientific Research (B) 16H03944, Japan Society for the Promotion of Science. We thank Natalia Saburova for Fig. . Finally, we would like to thank the referees for thoughtful comments that helped us to improve the manuscript.

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