The inverse spectral problem for transmission eigenvalues

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 η 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 η whenever 0