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.
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