On the inverse transmission eigenvalue problem with a piecewise W₂¹ refractive index

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∈ W²₂, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $ρ$ is a piecewise W¹₂ function. We prove that if ∫₀ᵇ √ρ(r) dr<b, then $ρ$ is uniquely determined by all special transmission eigenvalues; if ∫₀ᵇ √ρ(r) dr=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".