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A relation between the positive and negative spectra of elliptic operators

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Abstract

We study the spectral properties of pairs of operators \(-\Delta \pm V\) and show that if their negative spectra are discrete, then their essential spectra fill the positive semi-axis. Analogous statements are proved for more general operators of the form \(m(i\nabla )\pm V\) as well as for operators on the lattice \(\mathbb {Z}^d\).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Rowan Killip

  2. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Stanislav Molchanov & Oleg Safronov

  3. National Research University “Higher School of Economics”, Moscow, Russian Federation, 101000

    Stanislav Molchanov

Authors
  1. Rowan Killip
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  2. Stanislav Molchanov
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  3. Oleg Safronov
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Corresponding author

Correspondence to Oleg Safronov.

Additional information

R. K. was supported, in part, by NSF Grant DMS-1600942, M.S. was supported by NSF Grant DMS-1410547.

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Killip, R., Molchanov, S. & Safronov, O. A relation between the positive and negative spectra of elliptic operators. Lett Math Phys 107, 1799–1807 (2017). https://doi.org/10.1007/s11005-017-0970-y

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  • DOI: https://doi.org/10.1007/s11005-017-0970-y

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