Uniqueness and stability for inverse potential scattering with multiple wavenumbers

Abstract

This paper is concerned with the inverse scattering problem of determining a real-valued potential for the three-dimensional Schrödinger equation from multi-wavenumber data. Under suitable a priori bounds, we establish local increasing stability from near-field Dirichlet data on a fixed sphere for polynomially decaying potentials, and global increasing stability from full far-field amplitudes for exponentially decaying potentials. The stability estimates consist of a noise term and a remainder term, with the latter decreasing as the upper measured wavenumber increases. We also obtain global uniqueness from exact near-field data under weaker regularity assumptions. The analysis combines analytic continuation, weighted resolvent estimates, and half-ray identities whose differences at chord endpoints cancel the unknown exterior contribution.