# The Morse and Maslov indices for Schrödinger operators

@article{Latushkin2018TheMA,
title={The Morse and Maslov indices for Schr{\"o}dinger operators},
author={Yuri Latushkin and Selim Sukhtaiev and Alim Sukhtayev},
journal={Journal d'Analyse Math{\'e}matique},
year={2018},
volume={135},
pages={345-387}
}
• Published 6 November 2014
• Mathematics
• Journal d'Analyse Mathématique
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…
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## References

SHOWING 1-10 OF 43 REFERENCES
Counting spectrum via the Maslov index for one dimensional -periodic Schrödinger operators
• Mathematics
• 2015
We study the spectrum of the Schrodinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $\theta-$periodic boundary conditions. For two such operators,
Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients
• Mathematics
• 2006
The Maslov index is a topological property of periodic orbits of finite-dimensional Hamiltonian systems that is widely used in semiclassical quantization, quantum chaology, stability of waves and
A Morse Index Theorem for Elliptic Operators on Bounded Domains
• Mathematics
• 2014
Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ⊂ ℝ n is deformed. For a family of domains {Ω t } t∈[a, b] we prove that the Morse
Manifold decompositions and indices of Schr\"odinger operators
• Mathematics
• 2015
The Maslov index is used to compute the spectra of different boundary value problems for Schr\"{o}dinger operators on compact manifolds. The main result is a spectral decomposition formula for a
MULTI-DIMENSIONAL MORSE INDEX THEOREMS AND A SYMPLECTIC VIEW OF ELLIPTIC BOUNDARY VALUE PROBLEMS
• Mathematics
• 2011
Morse Index Theorems for elliptic boundary value problems in multi-dimensions are proved under various boundary conditions. The theorems work for star-shaped domains and are based on a new idea of
The Spectral Flow and the Maslov Index
• Mathematics
• 1995
exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of