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}
}
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… 
Counting spectrum via the Maslov index for one dimensional -periodic Schrödinger operators
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,
First-order asymptotic perturbation theory for extensions of symmetric operators
This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new
The Morse and Maslov indices for multidimensional Schr\"odinger operators with matrix-valued potentials
We study the Schr\"odinger operator $L=-\Delta+V$ on a star-shaped domain $\Omega$ in $\mathbb{R}^d$ with Lipschitz boundary $\partial\Omega$. The operator is equipped with quite general Dirichlet-
The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on [0, 1]
Working with general linear Hamiltonian systems on [0, 1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, we develop a general framework for relating
AN INDEX THEOREM FOR SCHRÖDINGER OPERATORS ON METRIC GRAPHS
We show that the spectral flow of a one-parameter family of Schrödinger operators on a metric graph is equal to the Maslov index of a path of Lagrangian subspaces describing the vertex conditions. In
Hadamard-type formulas via the Maslov form
Given a star-shaped bounded Lipschitz domain $${\Omega\subset{\mathbb{R}}^d}$$Ω⊂Rd, we consider the Schrödinger operator $${L_{\mathcal{G}}=-\Delta+V}$$LG=-Δ+V on $${\Omega}$$Ω and its restrictions

References

SHOWING 1-10 OF 43 REFERENCES
Counting spectrum via the Maslov index for one dimensional -periodic Schrödinger operators
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
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
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
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
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
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
...
...