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The Morse and Maslov indices for Schrödinger operators

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Abstract

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Missouri, Columbia, MO, 65211, USA

    Yuri Latushkin

  2. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Selim Sukhtaiev

  3. Department of Mathematics, Miami University, Oxford, OH, 45056, USA

    Alim Sukhtayev

Authors
  1. Yuri Latushkin
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  2. Selim Sukhtaiev
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  3. Alim Sukhtayev
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Corresponding author

Correspondence to Yuri Latushkin.

Additional information

Partially supported by the NSF grants DMS-1067929 and DMS-1710989, by the Simons Foundation, and by the Research Council and the Research Board of the University of Missouri.

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Latushkin, Y., Sukhtaiev, S. & Sukhtayev, A. The Morse and Maslov indices for Schrödinger operators. JAMA 135, 345–387 (2018). https://doi.org/10.1007/s11854-018-0043-x

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  • DOI: https://doi.org/10.1007/s11854-018-0043-x

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