Spectra of self-adjoint extensions and applications to solvable Schroedinger operators

Abstract

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.

81 pages, new references added, subsection 1.3 extended, typos corrected

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Cited by in corpus (86)

• Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
• On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set
• On the spectral properties of Dirac operators with electrostatic $\delta$-shell interactions
• A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
• Spectral Theory of Infinite Quantum Graphs
• Two-dimensional Dirac operators with singular interactions supported on closed curves
• One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results
• Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures
• Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary
• Eigenvalues of Robin Laplacians in infinite sectors
• First order operators and boundary triples
• Spectral estimates for resolvent differences of self-adjoint elliptic operators
• General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
• An example of unitary equivalence between self-adjoint extensions and their parameters
• Self-adjoint Dirac operators on domains in ${\mathbb{R}}^3$
• Formally self-adjoint quasi-differential operators and boundary value problems
• Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas
• Scattering on periodic metric graphs
• Spectral series of the Schrödinger operator with delta-potential on a three-dimensional spherically symmetric manifold
• The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
• Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions
• Edge States at Phase Boundaries and Their Stability
• Continuity of the integrated density of states on random length metric graphs
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Equilateral quantum graphs and boundary triples
• Spectral Estimates for Infinite Quantum Graphs
• M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems
• Quasi boundary triples and semibounded self-adjoint extensions
• Self-adjoint extensions and unitary operators on the boundary
• Self-adjoint indefinite Laplacians
• Levinson's theorem and higher degree traces for Aharonov-Bohm operators
• Unbounded quantum graphs with unbounded boundary conditions
• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• Moving Walls and Geometric Phases
• Spectral analysis of metric graphs and related spaces
• Localization on quantum graphs with random edge lengths
• New relations between discrete and continuous transition operators on (metric) graphs
• Spectral Analysis of a Two Body Problem with Zero Range Perturbation
• Localization on quantum graphs with random vertex couplings
• A self-adjointness criterion for the Schrödinger operator with infinitely many point interactions and its application to random operators
• Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations
• Self-adjoint Extensions of Restrictions
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• Dirac-like operators on the Hilbert space of differential forms on manifolds with boundaries
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• Bar codes of persistent cohomology and Arrhenius law for p-forms
• Absolutely Continuous Spectrum for Quantum Trees
• Role of anomalous symmetry in 0-$\pi$ qubits
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• Cantor spectra of magnetic chain graphs
• Spectrum of a family of spin-orbit coupled Hamiltonians with singular perturbation
• On the Number of Bound States of Point Interactions on Hyperbolic Manifolds
• Wave operators, similarity and dynamics for a class of Schroedinger operators with generic non-mixed interface conditions in 1D
• Laplacians on infinite graphs: discrete vs continuous
• Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps
• Scalar problems in junctions of rods and a plate. II. Self-adjoint extensions and simulation models
• Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions
• Zero modes of quantum graph Laplacians and an index theorem
• Perturbation determinants and trace formulas for singular perturbations
• A Perturbative Approach to the Tunneling Phenomena
• Self-adjoint extension for Maxwell-Chern-Simons model in long wavelength limit
• Embedded trace operator for infinite metric trees
• Scattering of particles bounded to an infinite planar curve
• Eigenvalue bracketing for discrete and metric graphs
• Krein-like extensions and the lower boundedness problem for elliptic operators
• Resolvent expansions on hybrid manifolds
• On the common Point Spectrum of pairs of Self-Adjoint Extensions
• A remark on the discriminant of Hill's equation and Herglotz functions
• Inverse scattering on the quantum graph -- Edge model for graphen
• Inverse scattering on the quantum graph for graphene
• The Jacobi operator and its Donoghue $m$-functions
• Numerical Solutions of the spectral problem for arbitrary self-adjoint extensions of the 1D Schroedinger equation
• Spectral and scattering theory for the Aharonov-Bohm operators
• A Mathematical Aspect of A Tunnel-Junction for Spintronic Qubit
• Boundary systems and (skew-)self-adjoint operators on infinite metric graphs
• Quantum Control at the Boundary: an application to quantum circuits
• Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core
• Boundary Data Maps for Schrodinger Operators on a Compact Interval
• Trace formulae for Schrödinger operators with singular interactions
• A class of nonlinear elliptic boundary value problems
• Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators
• A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains
• On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2
• A model for reducing the angulon operator
• Quantum Systems with jump-discontinuous mass. I