Spectral Theory and Analysis
Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland
1st Edition
Published on 29. March 2011
IX, 172 pages
978-3-7643-9994-8 (ISBN)
Description
This volume contains the proceedings of the OTAMP 2008 (Operator Theory, Analysis and Mathematical Physics) conference held at the Mathematical Research and Conference Center in Bedlewo near Poznan. It is composed of original research articles describing important results presented at the conference, some with extended review sections, as well as presentations by young researchers. Special sessions were devoted to random and quasi-periodic differential operators, orthogonal polynomials, Jacobi and CMV matrices, and quantum graphs. This volume also reflects new trends in spectral theory, where much emphasis is given to operators with magnetic fields and non-self-adjoint problems.
The book is geared towards scientists from advanced undergraduate students to researchers interested in the recent development on the borderline between operator theory and mathematical physics, especially spectral theory for Schrödinger operators and Jacobi matrices.
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Jan Janas | Pavel Kurasov | A. Laptev
Spectral Theory and Analysis
Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland
Book
03/2011
Birkhäuser
€106.99
Shipment within 10-15 days
Content
- 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
