JournalsjstVol. 7, No. 3pp. 633–658

Eigenvalue bounds for Schrödinger operators with complex potentials. II

  • Rupert L. Frank

    Caltech, Pasadena, USA and University of Munich, Germany
  • Barry Simon

    Caltech, Pasadena, USA
Eigenvalue bounds for Schrödinger operators with complex potentials. II cover

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Abstract

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator Δ+V-\Delta+V in L2(Rν)L^2(\mathbb R^\nu) with complex potential has absolute value at most a constant times Vγ+ν/2(γ+ν/2)/γ\|V\|_{\gamma+\nu/2}^{(\gamma+\nu/2)/\gamma} for 0<γν/20<\gamma\leq\nu/2 in dimension ν2\nu\geq 2. We prove this conjecture for radial potentials if 0<γ<ν/20<\gamma<\nu/2 and we 'almost disprove' it for general potentials if 1/2<γ<ν/21/2<\gamma<\nu/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.

Cite this article

Rupert L. Frank, Barry Simon, Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7 (2017), no. 3, pp. 633–658

DOI 10.4171/JST/173