On the inverse transmission eigenvalue problem with a piecewise W 2 1 refractive index

Abstract In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius b . Instead of the usual case of the refractive index ρ ∈ W 2 2 , by using singular Sturm–Liouville theory, we discuss the case when the refractive index ρ is a piecewise W 2 1 function. We prove that if ∫ 0 b ρ ( r ) d r < b , then ρ is uniquely determined by all special transmission eigenvalues; if ∫ 0 b ρ ( r ) d r = b , then all special transmission eigenvalues with some additional information can uniquely determine ρ . We also consider the mixed spectral problem and obtain that ρ is uniquely determined from partial information of ρ and the ‘almost real subspectrum’.