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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.

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.

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.

We consider Schrödinger operators = −Δ + in (Ω) where the domain Ω ⊂ ℝ and the potential = () are periodic with respect to the variable ∈ ℝ. We assume that Ω is unbounded with respect to the variable ∈ ℝ and that decays with respect to this variable. may contain a singular term supported on the boundary.

We develop a scattering theory for 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 is purely absolutely continuous and the scattering is complete.

We discuss the definition of a rank one singular perturbation of a non-self-adjoint operator in Hilbert space . Provided that the operator is a non-self-adjoint perturbation of a self-adjoint operator and that the spectrum of the operator 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 .

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.

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.

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.

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 (ℝ) by a finite number of point interactions, and the Schrödinger operator on the half-axis (0, ∞) in the Weyl limit circle case at infinity.

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.