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2007 | Buch

Finiteness of Eigenvalues of the Perturbed Dirac Operator Kapitel lesen Erstes Kapitel lesen

Operator Theory, Analysis and Mathematical Physics

herausgegeben von: Jan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko, Günter Stolz

Verlag: Birkhäuser Basel

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter
Finiteness of Eigenvalues of the Perturbed Dirac Operator
Abstract
Finiteness criteria are established for the point spectrum of the perturbed Dirac operator. The results are obtained by applying the direct methods of the perturbation theory of linear operators. The particular case of the Hamiltonian of a Dirac particle in an electromagnetic field is also considered.
Petru A. Cojuhari
A Mathematical Study of Quantum Revivals and Quantum Fidelity
Abstract
In this paper we present some results obtained recently, partly in collaboration with Didier Robert, about “quantum revivals” and “quantum fidelity”, mainly in the semiclassical framework. We also describe the exact properties of the quantum fidelity (also called Loschmidt echo) for the case of explicit quadratic plus inverse quadratic time-periodic Hamiltonians and establish that the quantum fidelity equals one for exactly the times where the classical fidelity is maximal.
Monique Combescure
On Relations Between Stable and Zeno Dynamics in a Leaky Graph Decay Model
Abstract
We use a caricature model of a system consisting of a quantum wire and a finite number of quantum dots, to discuss relation between the Zeno dynamics and the stable one which governs time evolution of the dot states in the absence of the wire. We analyze the weak coupling case and argue that the two time evolutions can differ significantly only at times comparable with the lifetime of the unstable system undisturbed by perpetual measurement.
Pavel Exner, Takashi Ichinose, Sylwia Kondej
On the Spectrum of Partially Periodic Operators
Abstract
We consider Schrödinger operators H = −Δ + V in L 2(Ω) where the domain Ω ⊂ ℝ + d+1 and the potential V = V (x, y) are periodic with respect to the variable x ∈ ℝd. We assume that Ω is unbounded with respect to the variable y ∈ ℝ and that V decays with respect to this variable. V may contain a singular term supported on the boundary.
We develop a scattering theory for H and present an approach to prove absence of singular continuous spectrum. Moreover, we show that certain repulsivity conditions on the potential and the boundary of Ω exclude the existence of surface states. In this case, the spectrum of H is purely absolutely continuous and the scattering is complete.
Rupert L. Frank, Roman G. Shterenberg
Functional Model for Singular Perturbations of Non-self-adjoint Operators
Abstract
We discuss the definition of a rank one singular perturbation of a non-self-adjoint operator L in Hilbert space H. Provided that the operator L is a non-self-adjoint perturbation of a self-adjoint operator A and that the spectrum of the operator L is absolutely continuous we are able to establish a concise resolvent formula for the singular perturbations of the class considered and to establish a model representation of it in the dilation space associated with the operator L.
Alexander V. Kiselev
Trace Formulas for Jacobi Operators in Connection with Scattering Theory for Quasi-Periodic Background
Abstract
We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein’s spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.
Johanna Michor, Gerald Teschl
Dirichlet-to-Neumann Techniques for the Plasma-waves in a Slot-diod
Abstract
Plasma waves in a slot-diod with governing electrodes are described by the linearized hydrodynamic equations. Separation of variables in the corresponding scattering problem is generally impossible. Under natural physical assumption we reduce the problem to the second order differential equation on the slot with an operator weight, defined by the Dirichlet-to-Neumann map of the three-dimensional Laplacian on the complement of the electrodes and the slot. The reduction is based on a formula for the Poisson map for the exterior Laplace Dirichlet problem on the complement of a few standard bodies in terms of the Poisson maps on the complement of each standard body.
Anna B. Mikhailova, Boris Pavlov, Victor I. Ryzhii
Inverse Spectral Problem for Quantum Graphs with Rationally Dependent Edges
Abstract
In this paper we study the problem of unique reconstruction of the quantum graphs. The idea is based on the trace formula which establishes the relation between the spectrum of Laplace operator and the set of periodic orbits, the number of edges and the total length of the graph. We analyse conditions under which is it possible to reconstruct simple graphs containing edges with rationally dependent lengths.
Marlena Nowaczyk
Functional Model of a Class of Non-selfadjoint Extensions of Symmetric Operators
Abstract
This paper offers the functional model of a class of non-selfadjoint extensions of a symmetric operator with equal deficiency indices. The explicit form of dilation of a dissipative extension is offered and the Sz.-Nagy-Foiaş model as developed by B. Pavlov is constructed. A variant of functional model for a non-selfadjoint non-dissipative extension is formulated. We illustrate the theory by two examples: singular perturbations of the Laplace operator in L 2(ℝ3) by a finite number of point interactions, and the Schrödinger operator on the half-axis (0, ∞) in the Weyl limit circle case at infinity.
Vladimir Ryzhov
Lyapunov Exponents at Anomalies of SL(2, ℝ)-actions
Abstract
Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible types. The associated invariant measure is calculated formally. For an anomaly of so-called second degree, it is given by the ground-state of a certain Fokker-Planck equation on the unit circle. The Lyapunov exponent is calculated to lowest order in perturbation theory with rigorous control of the error terms.
Hermann Schulz-Baldes
Uniform and Smooth Benzaid-Lutz Type Theorems and Applications to Jacobi Matrices
Abstract
Uniform and smooth asymptotics for the solutions of a parametric system of difference equations are obtained. These results are the uniform and smooth generalizations of the Benzaid-Lutz theorem (a Levinson type theorem for discrete linear systems) and are used to develop a technique for proving absence of accumulation points in the pure point spectrum of Jacobi matrices. The technique is illustrated by proving discreteness of the spectrum for a class of unbounded Jacobi operators.
Luis O. Silva
An Example of Spectral Phase Transition Phenomenon in a Class of Jacobi Matrices with Periodically Modulated Weights
Abstract
We consider self-adjoint unbounded Jacobi matrices with diagonal q n = n and weights λ n = c n n, where c n is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete. This constitutes an example of the spectral phase transition of the first order. We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (−∞; 1/2) or the interval (1/2; +∞) is covered by the absolutely continuous spectrum, the remainder of the spectrum being pure point. The proof is based on finding asymptotics of generalized eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate case, which constitutes yet another example of the spectral phase transition.
Sergey Simonov
On Connection Between Factorizations of Weighted Schur Functions and Invariant Subspaces
Abstract
We study operator-valued functions of weighted Schur classes over multiply-connected domains. There is a correspondence between functions of weighted Schur classes and so-called “conservative curved” systems introduced in the paper. In the unit disk case the fundamental relationship between invariant subspaces of the main operator of a conservative system and factorizations of the corresponding Schur class function (characteristic function) is well known. We extend this connection to weighted Schur classes. With this aim we develop new notions and constructions and make suitable changes in standard theory.
Alexey Tikhonov
Backmatter
Metadaten
Titel
Operator Theory, Analysis and Mathematical Physics
herausgegeben von
Jan Janas
Pavel Kurasov
Ari Laptev
Sergei Naboko
Günter Stolz
Copyright-Jahr
2007
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-7643-8135-6
Print ISBN
978-3-7643-8134-9
DOI
https://doi.org/10.1007/978-3-7643-8135-6

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