Inverse Resonance Problem in Scattering Theory

• Inverse Resonance Problem is a central topic in spectral theory that determines operators from their resonances, bridging analytic functions with physical observables.
• It utilizes scattering data, such as Jost functions and reflection coefficients, to reconstruct potentials and operator parameters via polynomial interpolation and Riemann surface analysis.
• Applications span quantum mechanics, geophysical inversions, and numerical reconstruction, offering insights into stability, uniqueness, and explicit inversion formulas.

The inverse resonance problem is a central topic in spectral and scattering theory, addressing the determination of an operator or potential from knowledge of its resonances—complex frequencies where the system exhibits enhanced response. Resonances generalize the concept of eigenvalues to non-selfadjoint or non-compact settings and correspond to the poles of analytically continued scattering or resolvent matrices. The problem connects the analytic structure of entire functions (Jost, scattering, or transmission functions) with physical observables and underpins reconstruction and uniqueness theorems for operators in both continuous and discrete frameworks.

1. Mathematical Formulation and Scattering Data

Inverse resonance problems originate in one-dimensional scattering theory for both continuous operators (e.g., Schrödinger, Dirac) and discrete analogs (Jacobi/CMV operators). The central objects are the scattering matrix $S(k)$ and its entries, which are rational or meromorphic functions of the spectral parameter $k$ or $z$. Scattering matrix entries are typically expressed via Fourier transforms or Wronskians of Jost functions. For Jacobi operators perturbed finitely on $\mathbb{Z}$, the central equations and scattering data are as follows:

• Periodic Jacobi Operator: The background operator $H^0$ acts as

$$(H^0y)_n = a_n^0 y_{n-1} + a_n^0 y_{n+1} + b_n^0 y_n$$

where $(a_n^0, b_n^0)$ are $q$-periodic sequences.

• Perturbation: Finitely supported changes $u_n$, $v_n$ define $a_n = a_n^0 + u_n$, $b_n = b_n^0 + v_n$ non-zero on a finite interval.
• Scattering Data: The pair $(\widehat{w}(X), f_+(X))$

$\widehat{w}(X) = 2i\sin(q x(X))(1+A(X)) - J(X)$

where $A(X)$ and $J(X)$ are real polynomials encoding the spectral perturbation. The Jost function $f_+$ solves the perturbed difference equation with prescribed asymptotics.

• Resonance Definition: Resonances are zeros of the Jost function or certain numerators/denominators of the reflection/transmission coefficients located off the real or “physical” axis (often on a nonphysical sheet of an associated Riemann surface). For the Schrödinger operator with compactly supported $q(x)$, the Jost function $\psi(z) = y(0, z)$, with $y(x,z)$ the Jost solution, is entire and its zeros in $\Im z < 0$ are resonances.
• Reflection/Transmission via Jost Functions: For Jacobi/Schrödinger/CMV models, meromorphic entries of the scattering matrix are ratios involving entire functions whose zeros determine resonance data. For example, in Jacobi or CMV settings, zeros of the Jost function (or related reflection coefficient numerators) are in bijection with resonances.

2. Uniqueness, Characterization, and Reconstruction

Numerous frameworks provide one-to-one correspondences between scattering data (including resonance locations) and underlying operator parameters, subject to normalization or translation ambiguities and support/decay assumptions. For periodic Jacobi operators with finitely supported perturbations:

• Uniqueness Theorem: The mapping from finitely supported perturbations $(u, v)$ to the pair $(\widehat{w}, f_+)$ (scattering data) is bijective when $v$ is in the admissible set corresponding to the support class (Theorem 1.2 in (Iantchenko et al., 2011)).
• Reconstruction: The explicit formulas for $\widehat{w}(X)$ in terms of polynomials $A(X)$ and $J(X)$ and their recovery from resonance data (zeros of $\widehat{w}$, zeros of $f_+$, and zeros of $R_+(\lambda)+1$) enable the reconstruction of the operator. Techniques such as polynomial interpolation, analysis on a two-sheeted Riemann surface, and discrete Gel'fand–Levitan–Marchenko equations realize this algorithmically.
• Spectral Measure Characterization: For periodic or eventually periodic Jacobi operators, one can write the spectral measure as

$$d\mu(x) = \frac{\sqrt{|r(x)|}}{|a(x)|} \, dx + \sum_j w_j \delta_{E_j}(x),$$

with $r(x)$ encoding band edges and $a(z)$ an analytic function whose zeros (partitioned between the physical and unphysical sheets) correspond to eigenvalues and resonances. Necessary and sufficient interlacing and regularity conditions (e.g., "oddly interlaced" singularities) fully characterize realizable resonance/eigenvalue sets (Kozhan, 2012).

• Cartwright Class and Hadamard Factorization: For operators with super-exponential or compactly supported perturbations, the relevant entire functions (e.g., the Jost function) admit Hadamard factorizations, and the resonance set uniquely determines these functions, and hence the potential or Jacobi coefficients.

3. Stability Analysis

The stability of the inverse resonance problem quantifies how small perturbations in resonances (e.g., errors in measured or computed zeros of scattering functions) affect the reconstructed operator.

• Quantitative Stability Estimates: For compactly supported potentials, the error $\delta(r)$ in the recovered potential from resonances in $|z|<r$ satisfies explicit decay rates $\delta(r) \leq C r^{-\alpha}$. The constants depend on a priori bounds on potential magnitude and support (Geynts et al., 2019).
• Stability for Jacobi/CMV Operators: If zeros and poles of the left (or right) reflection coefficient of two finitely supported Jacobi operators are $\epsilon$-close in a disk of radius $R$, the pointwise difference in coefficients is bounded by $C(\epsilon + 1/R)$ (Bledsoe, 2012). In the CMV (unitary) case, analogous estimates control the difference in Verblunsky coefficients as a function of proximity of resonance sets (Shterenberg et al., 2013).
• Stability via Value Distribution Theory: When resonance zeros are asymptotically regularly distributed (e.g., under Cartwright's theorem), natural stability emerges from the direct connection between the width of the Fourier indicator diagram for the potential and the density/distribution of zeros, implying that the support size of the potential governs the regularity and density of the resonances (Chen, 14 Aug 2025).

4. Inverse Resonance in Discrete and Periodic Settings

Discrete and periodic operators introduce additional structural elements in the inverse resonance problem:

• Two-Sheeted Riemann Surface: Scattering and resonance analysis for periodic backgrounds occurs naturally on a two-sheeted Riemann surface constructed from the quasi-momentum mapping, with bound states and resonances lying on different sheets. Analytic functions ($\widehat{w}$, $f_+$) are defined on this surface, and their zero patterns encode all scattering and spectral information (Iantchenko et al., 2011).
• Systematic Handling of Multiplicities and Forbidden Regions: For Jacobi operators on the half-lattice with finitely supported perturbations, the characteristic polynomial encoding resonances satisfies explicit conditions on allowable multiplicities and the forbidden domains for non-real or highly degenerate resonances (Korotyaev et al., 2022). This discretely mirrors forbidden domains for resonance locations in continuous settings.
• Algorithmic Approaches: Discrete resonance problems can often be solved with finite-algebraic and interpolation methods (rather than integral equations), based on the roots of the Jost function polynomial and the recovery of coefficients via inverse eigenvalue theory (Korotyaev et al., 2022).

5. Generalizations and Applications

The inverse resonance problem has wide-ranging implications across mathematical physics, geometric analysis, and applied fields:

• Spectral Geometry and Manifold Problems: Inverse resonance uniqueness results extend to Laplacians on bounded and exterior domains, Laplace–Beltrami operators on compact or hyperbolic manifolds, and scattering from obstacles. In geometric contexts, resonance data, via trace and Poisson formulas, recover geometric invariants, yielding uniqueness and rigidity results (Datchev et al., 2011).
• Physical Reconstruction: Applications include quantum mechanics (where resonances correspond to metastable or decaying states), wave propagation in periodic/discrete media, and geophysical inverse problems (such as reconstructing elastic parameters from scattering data) (Sottile, 2022).
• Numerical Reconstruction: Stability results justify numerical approaches that reconstruct potentials from finite or noisy resonance data, enabling rigorous error control in algorithms based on analytic or polynomial representations of scattering functions.
• Extensions: Ongoing research explores stability under more general perturbations, robustness to missing or incomplete resonance data, and adapts methodologies to operators with different background structures (e.g., massless Dirac, energy-dependent, or doubly infinite systems).

6. Advanced Analytical Techniques

Solving the inverse resonance problem requires integrating complex analysis (entire and meromorphic function theory), Riemann surface analysis, and operator theory:

• Transformation and Marchenko Operators: The Gel'fand–Levitan–Marchenko equation and transformation operators translate between spectral data and operator coefficients, both for continuum and discrete systems. These equations can take integral form (in Schrödinger settings) or discrete summation form (in Jacobi/CMV settings), with kernels reconstructed from resonance data via inverse Fourier transform or polynomial interpolation (Iantchenko et al., 2011).
• Asymptotics and Zero Distribution: The density and angular distribution of resonance zeros follow asymptotic laws dictated by the support and smoothness of the potential, which are captured through growth indicators and classical theorems (Jensen, Cartwright, Levinson), playing a critical role in both uniqueness and stability arguments.
• Explicit Reconstruction Formulas: For instance, with known resonance sets and norming constants, the potential can be reconstructed via the Marchenko kernel or Gelfand–Levitan equation, using explicit formulas for the transformation kernel in terms of the scattering data (Sottile, 2022).

7. Summary Table of Central Constructs

Operator Class	Resonance Definition	Inversion/Uniqueness Mechanism
Periodic Jacobi w/f.s.p.	Zeros of $\widehat{w}(X)$ on Riemann surface	Bi-unique correspondence $(u,v) \leftrightarrow (\widehat{w}, f_+)$ via polynomial interpolation and Marchenko equation (Iantchenko et al., 2011)
Jacobi, discrete	Zeros of Jost polynomial outside unit disk	Algebraic inverse via finite eigenvalue problem; uniqueness and forbidden domain analysis (Korotyaev et al., 2022)
Continuous Schrödinger	Zeros of Jost function in $\mathbb{C}_-$	Hadamard factorization and Cartwright theory; stability in normed spaces (Geynts et al., 2019, Chen, 14 Aug 2025)
CMV (unitary, discrete)	Zeros of Jost function (growth order zero)	Schur algorithm: reconstruct Verblunsky coefficients from Weyl–Titchmarsh function (Shterenberg et al., 2013)

• For periodic Jacobi operators with finitely supported perturbations, the principal results and reconstruction algorithms are detailed in "Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem" (Iantchenko et al., 2011).
• Complete characterizations, including stability and the spectral measure structure, are in "Spectral and resonance problem for perturbations of periodic Jacobi operators" (Kozhan, 2012).
• For the stability of resonance-based inversion in continuous one-dimensional settings, see "Stability in the inverse resonance problem for the Schrödinger operator" (Geynts et al., 2019) and "Stability of Inverse Resonance Problem on the Half Line" (Chen, 14 Aug 2025).
• Discrete inverse resonance theory, including forbidden domains and multiplicity constraints, is summarized in "Inverse problems for Jacobi operators with finitely supported perturbations" (Korotyaev et al., 2022).
• CMV operator stability results and analytic inversion via the Schur algorithm are in (Shterenberg et al., 2013).