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Sabine Bögli

Schrödinger operator with non-zero accumulation points of complex eigenvalues

Abstract: In the 1960s Pavlov studied Schrödinger operators on the half-line with potentials that decay at infinity, subject to Robin boundary conditions at the endpoint. Using inverse spectral theory, he proved the existence of a real potential and a non-selfadjoint boundary condition so that the Schrödinger operator has infinitely many non-real eigenvalues that accumulate at an arbitrary prescribed point of the essential spectrum (the positive half-line). Since then, it has been an open question whether these results can be modified so that the non-selfadjointness is not coming from the boundary conditions but from a non-real potential.

In this talk we consider Schrödinger operators on the whole Euclidean space (of arbitrary dimension) or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schrödinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum.

The talk is based on a recent publication in Communications in Mathematical Physics