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Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator in with complex potential has absolute value at most a constant times for in dimension . We prove this conjecture for radial potentials if and we 'almost disprove' it for general potentials if . In addition, we prove various bounds that hold, in particular, for positive eigenvalues.
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Rupert L. Frank, Barry Simon, Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7 (2017), no. 3, pp. 633–658DOI 10.4171/JST/173