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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 $$ \left[ b,\pi \right] $$ b , π with $$b\in \left( d,\pi \right] $$ b ∈ d , π or $$b=d$$ b = d , then $$h,\,\beta ,\,\gamma $$ h , β , γ and q on $$\left[ 0,\pi \right] $$ 0 , π 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 $$b\in \left( 0,d\right) $$ b ∈ 0 , d , a similar statement holds if $$\beta ,\,\gamma $$ β , γ 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.

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