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The inverse spectral problem for transmission eigenvalues. (English) Zbl 1372.35361

Summary: In this paper, we consider the inverse medium problem of determining the spherically stratified index of refraction \(n(r)\) from given spectral data. We begin by introducing a modified transmission eigenvalue problem depending on a parameter \(\eta\) and an associated modified far field operator. We prove that this operator is injective with dense range provided that \(k\) is not a modified transmission eigenvalue, and we show that \(n(r)\) is uniquely determined by the modified transmission eigenvalues corresponding to \(\eta\) whenever \(0<n(r)<\eta^2\) for \(0\leqslant r\leqslant 1\).

MSC:

35R30 Inverse problems for PDEs
35P05 General topics in linear spectral theory for PDEs
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