On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ 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 $\rho \in W_2^2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $\rho$ is a piecewise $W_2^1$ function. We prove that if ${\int }_0^b\sqrt{\rho (r)}dr<b$, then $\rho$ is uniquely determined by all special transmission eigenvalues; if ${\int }_0^b\sqrt{\rho (r)}dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $\rho$. We also consider the mixed spectral problem and obtain that $\rho$ is uniquely determined from partial information of $\rho$ and the ``almost real subspectrum".