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

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 Schrödinger equation.

30 pages, no figures

Cited by in corpus (18)

• Reconstruction of the wave speed from transmission eigenvalues for the spherically-symmetric variable-speed wave equation
• On a regularization approach to the inverse transmission eigenvalue problem
• Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem
• A CMV--based eigensolver for companion matrices
• Application of machine learning regression models to inverse eigenvalue problems
• Inverse problem with transmission eigenvalues for the discrete Schrödinger equation
• Transmission eigenvalues for the selfadjoint Schrödinger operator on the half line
• On the direct and inverse transmission eigenvalue problems for the Schrodinger operator on the half line
• Fast and backward stable computation of roots of polynomials, Part II: backward error analysis; companion matrix and companion pencil
• A Global Uniqueness on Spherically Stratified Dielectric Medium in Time-Harmonic Maxwell Equation with Interior Transmission Eigenvalues
• An Uniqueness Result on Spherically Stratified Media in Constant Absorbing Background with Interior Transmission Eigenvalues
• An Uniqueness Result on Spherically Stratified Media with Interior Transmission Eigenvalues
• Some Inverse Spectral Results in Exterior Transmission Problem
• A Near-Field Basis in Radially Symmetric Interior Transmission Problem
• A Fixed Energy Fixed Angle Inverse Uniqueness in Interior Transmission Problem
• An Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Penetrable Simple Domains
• A Proof of Schiffer's Conjecture in Starlike Domain by Far-Field Patterns
• A Density Theorem on the Eigenvalues of Spherically Symmetric Interior Transmission Problem in Absorbing Medium