Lippmann-Schwinger-Lanczos approach for inverse scattering problem of Schrodinger equation in the resonance frequency domain

TL;DR

The paper addresses inverse scattering for the Schrödinger equation using data in the resonance frequency domain. It combines the Lippmann-Schwinger framework with data-driven reduced-order models, constructed via the Loewner framework and Lanczos-based rational Krylov reductions, to enable direct linear imaging of the potential from measurements. A Weyl-law–based sampling strategy selects resonance frequencies to balance accuracy and computational load. In 1D noiseless tests, the resonance-domain LSL approach yields sharper reconstructions than the diffusive-domain method, accurately recovering smooth and, to a lesser extent, discontinuous potentials (with Gibbs artifacts near jumps). The work suggests a scalable pathway for high-resolution inverse scattering, with future work focusing on multi-dimensional problems and noisy data.

Abstract

Reconstructions of potential in Schrodinger equation with data in the diffusion frequency domain have been successfully obtained within Lippmann-Schwinger-Lanczos (LSL) approach, however limited resolution away from the sensor positions resulted in rather blurry images. To improve the reconstructions, in this work we extended the applicability of the approach to the data in the resonance frequency domain. We proposed a specific data sampling according to Weyl's law that allows us to obtain sharp images without oversampling and overwhelming computational complexity. Numerical results presented at the end illustrate the performance of the algorithm.

Figures (2)

• Figure 1: $\mathbf{u}$ and $u$ almost coincide while $u^0$ is totally different (see (a) plot for $f=4$). True continuous potential $p$, its reconstructions $p_{Born}$ via Born and $p_{LSL}$ via LSL for $f=3$ see (b), $f=4$ (see (c) and $f=5$ (see (d))
• Figure 2: True discontinuous potential $p$, its reconstructions $p_{Born}$ via Born and $p_{LSL}$ via LSL