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Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 345–387 | Cite as

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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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  • Yuri Latushkin
    • 1
  • Selim Sukhtaiev
    • 2
  • Alim Sukhtayev
    • 3
  1. 1.Department of MathematicsThe University of MissouriColumbiaUSA
  2. 2.Department of MathematicsRice UniversityHoustonUSA
  3. 3.Department of MathematicsMiami UniversityOxfordUSA

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