Spectral Theory and Analysis

Name: Spectral Theory and Analysis | Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland
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Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland

Jan Janas Pavel Kurasov A. Laptev Sergei Naboko Günter Stolz(Herausgeber*in)

Birkhäuser (Verlag)

1. Auflage

Erschienen am 29. März 2011

IX, 172 Seiten

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Intro
Spectral Theory and Analysis
Contents
Introduction
Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients
1. Introduction
2. Definitions and preliminary results
3. Floquet transform in Hm(Rd) and H-m(Rd)
4. Floquet-Bloch theory
5. Equality of the H-m and L2-spectra
References
Representations of Compact Linear Operators in Banach Spaces and Nonlinear Eigenvalue Problems II
1. Introduction
2. The representation theorems
3. Applications
References
A Sharp Bound on Eigenvalues of Schrödinger Operators on the Half-line with Complex-valued Potentials
1. Introduction and main result
2. Proofs
References
Zero-range Model of p-scattering by a Potential Well
1. Introduction
2. P-scattered waves and boundary conditions
3. Space Ha and Operator Aa
4. Proof of Theorem 1
5. Construction of the energy operator
6. Possible Generalizations
References
The Similarity Problem for Non-selfadjoint Operators with Absolutely Continuous Spectrum: Restrictions to Spectral Subspaces
1. Preliminaries
2. The functional model
3. Similarity problem for additive non-selfadjoint perturbations with absolutely continuous spectrum
4. Application: Friedrichs model operator
Acknowledgement
References
Generalized Eigenfunctions and Spectral Theory for Strongly Local Dirichlet Forms
Introduction
1. Strongly local Dirichlet forms
Dirichlet forms
Capacity
Strong locality and the energy measure
Irreducibility
Assumptions
2. Measure perturbations
3. Weak solutions
4. Positive weak solutions and the infimum of the spectrum
Ground state transform and consequences
Harnack principles and existence of positive solutions below the spectrum
Characterizing the infimum of the spectrum
5. Weak solutions and spectrum
A Weyl type criterion
A Caccioppoli type inequality
A 1/2 Shnol type result: How suitably bounded solutions force spectrum
A 1/2 Shnol type result: How spectrum forces suitably bounded generalized eigenfunctions.
A Shnol type result: Characterizing the spectrum by subexponentially bounded solutions
6. Examples and applications
Hamiltonians with singular interactions
Quantum graphs
Applications
References
Trace Formulas for Schrödinger Operators in Connection with Scattering Theory for Finite-gap Backgrounds
1. Introduction
2. Notation
3. Asymptotics of Jost solutions
4. Connections with Krein's spectral shift theory and trace formulas
5. The transmission coefficient
6. Conserved quantities of the KdV hierarchy
Acknowledgment
References
Inner-outer Factorization for Weighted Schur Class Functions and Corresponding Invariant Subspaces
0. Introduction
1. Factorizations and invariant subspaces
2. Inner-outer factorization
3. Description of outer functions and corresponding invariant subspaces
References
Eigenvalue Asymptotics for Magnetic Fields and Degenerate Potentials
1. Introduction
2. Degenerate potentials
2.1. The Tauberian approach
2.2. The min-max approach
3. Magnetic bottles
3.1. General setting
3.2. The Euclidean case
3.2.1. The results.
3.2.2. The Dirichlet problem in a cube for a constant magnetic field.
3.2.3. A subdivision of Rd into appropriate cubes.
3.3. The hyperbolic half-plane
3.3.1. The setup.
3.3.2. The Maass Laplacian.
3.3.3. Non-Weyl-type asymptotics (high energy).
3.3.4. Outline of the proof.
3.4. Geometrically finite hyperbolic surfaces
3.4.1. Introduction.
3.4.2. Definition.
3.4.3. Assumptions on the magnetic field.
3.4.4. Asymptotics for large energies.
4. A Neumann problem with magnetic field
4.1. A problem arising from super-conductivity
4.2. The spectrum in the case of the half-space, for a constant field and for h = 1
4.3. Non-Weyl-type asymptotics when the field is nearly tangent to the boundary
5. A problem of magnetic bottle in classical mechanics
5.1. The Lorentz equation
5.2. Adiabatic invariants
5.3. Bounded trajectories
5.3.1. The Hamiltonian in cylindrical coordinates.
5.3.2. The reduced Hamiltonian and the magnetic field lines.
5.3.3. Action variables.
6. Open problems and conclusion
References
List of Participants

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