The Morse and Maslov indices for Schrödinger operators

被引:0
|
作者
Yuri Latushkin
Selim Sukhtaiev
Alim Sukhtayev
机构
[1] The University of Missouri,Department of Mathematics
[2] Rice University,Department of Mathematics
[3] Miami University,Department of Mathematics
来源
Journal d'Analyse Mathématique | 2018年 / 135卷
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摘要
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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页码:345 / 387
页数:42
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