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We consider Schr\"odinger operators in $\mathbb R^d$ with complex potentialssupported on a hyperplane and show that all eigenvalues lie in a disk in thecomplex plane with radius bounded in terms of the $L^p$ norm of the potentialwith $d-1<p\leq d$. We also prove bounds on sums of powers of eigenvalues.

Eigenvalues of Schr\"odinger operators with complex surface potentials