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Frank, Rupert L. and Simon, Barry (2017): Eigenvalue bounds for Schrödinger operators with complex potentials. II. In: Journal of Spectral Theory, Vol. 7, No. 3: pp. 633-658 [PDF, 284kB]

Abstract

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator −Δ+Vin L2(Rν) with complex potential has absolute value at most a constant times ∥V∥(γ+ν/2)/γγ+ν/2 for 0<γ≤ν/2 in dimension ν≥2. We prove this conjecture for radial potentials if 0<γ<ν/2 and we 'almost disprove' it for general potentials if 1/2<γ<ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.

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