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

Abstract

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.

References in corpus (2)

• Uniqueness for Inverse Sturm-Liouville Problems with a Finite Number of Transmission Conditions
• Inverse spectral problems for the Sturm-Liouville operator with discontinuity