Transmission eigenvalues and thermoacoustic tomography

TL;DR

This paper explores how the spectrum of the interior transmission problem relates to determining acoustic properties in thermoacoustic imaging, showing spectral sparsity implies certain range conditions, especially in odd dimensions.

Contribution

It establishes a connection between transmission eigenvalues and range separation in thermoacoustic operators, providing new insights for radially symmetric non-trapping sound speeds.

Findings

Transmission spectrum sparsity implies range separation.

In odd dimensions ≥3, the spectrum for radially symmetric speeds is countable.

Ranges of thermoacoustic maps have only trivial intersection.

Abstract

The spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. Under a non-trapping hypothesis, we show that sparsity of the interior transmission spectrum implies a range separation condition for the thermoacoustic operator. In odd dimension greater than or equal to three, we prove that the transmission spectrum for a pair of radially symmetric non-trapping sound speeds is countable, and conclude that the ranges of the associated thermoacoustic maps have only trivial intersection.