Shnol-type theorem for the Agmon ground state

  • Siegfried Beckus

    Universität Potsdam, Germany
  • Yehuda Pinchover

    Technion - Israel Institute of Technology, Haifa, Israel
Shnol-type theorem for the Agmon ground state cover
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Abstract

Let be a Schrödinger operator defined on a noncompact Riemannian manifold , and let . Suppose that the operator is critical in , and let be the corresponding Agmon ground state. We prove that if is a generalized eigenfunction of satisfying in , then the corresponding eigenvalue is in the spectrum of . The conclusion also holds true if for some the operator admits a positive solution in , and in , where is a positive solution of minimal growth in a neighborhood of infinity in . Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

Cite this article

Siegfried Beckus, Yehuda Pinchover, Shnol-type theorem for the Agmon ground state. J. Spectr. Theory 10 (2020), no. 2, pp. 355–377

DOI 10.4171/JST/296