Inverse Stability on Scattering Source

• Inverse stability on scattering source is defined as the quantitative link between small perturbations in scattering data and the stable recovery of potentials.
• The approach combines spectral theory, complex analysis, and Fourier techniques to relate shifts in resonance zeros to variations in compactly supported potentials.
• Practical inversion exploits the robustness of resonances, where minimal changes in resonance locations ensure accurate and stable reconstruction of the underlying source.

Inverse stability on scattering source describes the relationship between measured scattering data and the stably recoverable properties of a physical scatterer or source. This concept encompasses quantitative estimates linking perturbations in the observed fields or resonance data to the corresponding perturbations in constitutive parameters such as potentials, impedances, or source distributions. In one-dimensional scattering theory, particularly the inverse resonance problem, inverse stability centers on reconstructing the compactly supported perturbation (such as a potential) from the zeros (resonances) of certain meromorphic or entire functions, typically Fourier transforms of wavefields. The mathematical underpinnings involve a deep interplay between spectral theory, complex analysis, perturbation theory of resonances, and value distribution theory.

1. Formulation of the Inverse Resonance Problem

In the context of one-dimensional scattering governed by the Schrödinger equation

$$\psi''(k, x) + k^2 \psi(k, x) = V(x) \psi(k, x), \quad x \in \mathbb{R},$$

the scattering data is encapsulated in the scattering matrix $S(k)$, whose entries are meromorphic functions in $k$. These entries are constructed using Jost solutions and their Fourier transforms: $$\phi_+(x, k) \sim \begin{cases} e^{ikx}, & x \rightarrow +\infty, \ [\hat{X}(k)/ik] e^{ikx} + [\hat{Y}(-k)/ik] e^{-ikx}, & x \rightarrow -\infty, \end{cases}$$ with a corresponding formulation for $\phi_-(x, k)$. The transmission coefficient is $T(k) = (ik)/\hat{X}(k)$. Resonances are defined as the zeros of $\hat{X}(k)$, i.e., the points in the complex $k$-plane where the analytic continuation of the scattering solution fails to be asymptotically outgoing. Resonances, along with bound states and their multiplicities, form the set $\Sigma$ that determines the inverse problem.

The foundational result is that knowledge of the resonance set provides, under suitable conditions, sufficient information to (stably) reconstruct the potential $V(x)$.

2. Meromorphic Structure and Fourier Analysis

Each entry of $S(k)$, including $T(k)$, $R(k)$, and $L(k)$, is expressed as a quotient of meromorphic functions: $$S(k) = \begin{bmatrix} T(k) & R(k) \ L(k) & T(k) \end{bmatrix},$$ where $T(k) = (ik)/\hat{X}(k)$ and $R(k)$, $L(k)$ are ratios involving $\hat{Y}(k)$ and $\hat{X}(k)$. These functions are analytic except at isolated poles determined by the zeros of $\hat{X}(k)$.

A central analytic tool is the representation of resonances as zeros of entire functions linked to the Fourier transform of $V(x)$ and related quantities. For instance, Froese’s theorem relates the collection of resonances asymptotically to the zero set of products such as $\hat{V}(2k)\hat{V}(-2k)$. The entire and meromorphic structure is essential for employing complex analytic techniques in the estimation and stability investigation.

3. Quantitative Stability via Resonance Perturbation

Quantitative stability is established by analyzing how small perturbations in the resonance set $\Sigma = \{z_n\}$ reflect in small (typically multiplicative) perturbations of the potential $V(x)$. For compactly supported or super-exponentially decaying potentials, the Fourier transform $\hat{V}(k)$ is entire of finite exponential type. Classical theorems from complex analysis, such as Cartwright's theorem, link the type and indicator diagram of $\hat{V}$ (determined by the support size of $V$) to the distribution and density of its zeros. Dickson's counting theorems for exponential polynomials provide explicit bounds for the number and location of resonances.

The paper proves a fundamental result: if the resonance zeros $\{\sigma_j^1\}$ and $\{\sigma_j^2\}$ corresponding to potentials $V^1$ and $V^2$ satisfy

$0 \leq \sup_j |\sigma_j^1 - \sigma_j^2| < \delta,$

then for any prescribed $\epsilon > 0$, there exists $\delta = \delta(\epsilon)$ such that

$$\left| \frac{V^1(x)}{V^2(x)} - 1 \right| < \epsilon,$$

for all $x$. This establishes that the potential (scattering source) is stable with respect to small perturbations of the resonance set.

4. Relation Between Potential Support and Zero Distribution

The spatial support (or effective width) of the potential $V(x)$ is intimately connected to the growth and zero distribution of the associated Fourier transforms. Larger support corresponds to larger exponential type and a wider indicator function $h_f(\theta) = \sigma |\sin \theta|$, which in turn determines the angular distribution of zeros according to Cartwright's theory.

This connection is also manifest in the explicit product expansions for $\hat{V}(2k)\hat{V}(-2k)$, where the exponential decay or support size is encoded in the exponential factors and the zero set. This structural relationship is used in the paper to derive stability estimates: the distribution of resonances provides direct information about the spatial characteristics of the scattering source.

5. Perturbation Theory and Value Distribution

Perturbation theory of resonances quantifies how perturbations of the potential lead to perturbations of the corresponding zeros of the Fourier transforms. When the zero sets for two potentials are close in the supremum norm (i.e., resonances shift by at most $\delta$), analytic factorization and argument principle techniques yield explicit bounds on the changes in the potential.

Further, value distribution theory (especially Cartwright's and related results) justifies "natural stability" for regularly distributed resonance sets. In particular, for entire functions of exponential type with regularly spaced zeros corresponding to physical scattering data, the inversion map from the resonance data to the source parameter is robust: small errors in most resonance locations only induce small changes in the potential.

6. Implications for Practical Inversion

These mathematical results have significant implications for the practical recovery of scattering sources in inverse problems:

• If resonance data are measured to sufficient completeness and accuracy, the underlying potential $V(x)$ can be determined with a controlled error, assuming regular zero distribution and compact support.
• The size of the potential's support is implicitly encoded in the asymptotic density of resonances, allowing for support determination from spectral data alone.
• The inverse resonance approach via zeros of Fourier transforms provides a pathway for stability analysis that avoids the need for full knowledge of the reflection/transmission coefficients in the physical domain.
• Practical inversion algorithms can exploit these stability mechanisms, leveraging the connection between resonance proximity and reconstruction accuracy, especially in measurements dominated by regular distributions of resonances.

7. Summary Table: Key Quantitative Relations

Concept	Mathematical Representation	Stability Principle
Resonance Set (Zeros)	$\{\sigma_j\}$: zeros of $\hat{X}(k)$, $\hat{V}(2k)$	$\sup_j |\sigma_j^1 - \sigma_j^2| < \delta \implies$
Transmission Coefficient	$T(k) = (ik)/\hat{X}(k)$	$|V^1(x)/V^2(x) - 1| < \epsilon$ if zeros are close
Potential Support/Zero Density	Indicator function $h_f(\theta)$; Cartwright's theory	Support size determines asymptotic resonances density
Regularly Distributed Resonances	Asymptotically linear counting of zeros	"Natural" stability via value distribution theory

In conclusion, inverse stability on scattering source in the context of one-dimensional resonance problems is governed by the analytic structure of the scattering matrix, the entire function nature of related Fourier transforms, and the value distribution of their zeros. Perturbation theory and entire function theory establish that sufficiently regular, closely distributed resonance data enable a robust and quantitatively stable reconstruction of the underlying source, with the spatial support and fine structure of the potential reflected in the distribution of scattering resonances (Chen, 14 Aug 2025).