Inverse spectral problems for non-self-adjoint Sturm-Liouville operators with discontinuous boundary conditions

This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $\left[ b,\pi \right]$ with $b\in \left( d,\pi \right]$ or $b=d$, then $h,$ $\beta ,$ $\gamma \$and $q$ on $\left[ 0,\pi \right] \$can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $b\in \left( 0,d\right) ,$ a similar statement holds if $\beta ,$ $\gamma \$are also known a priori. Moreover, if $q$ satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl $m$-function to solve the problem of missing eigenvalues and norming constants.