The uniqueness in the inverse problem for transmission eigenvalues for the spherically-symmetric variable-speed wave equation

TL;DR

This paper proves the unique determination of a spherically symmetric wave speed within a sphere from transmission eigenvalues and their multiplicities, under certain integral conditions, with extensions to related Schrödinger equations.

Contribution

It establishes new uniqueness results for recovering wave speed from transmission eigenvalues, including conditions involving multiplicities and additional data when the integral equals the radius.

Findings

Unique recovery of wave speed when integral of 1/v is less than b.

Additional data needed for uniqueness when the integral equals b.

Extension of results to Schrödinger equation eigenvalues.

Abstract

The recovery of a spherically-symmetric wave speed $v$ is considered in a bounded spherical region of radius $b$ from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of $1/v$ on the interval $[0,b]$ is less than $b,$ assuming that there exists at least one $v$ corresponding to the data, it is shown that $v$ is uniquely determined by the data consisting of such transmission eigenvalues and their "multiplicities," where the "multiplicity" is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to $b,$ the unique recovery is obtained when the data contains one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with "multiplicities" for a related Schr\"odinger equation.