Related papers: Trace Formulas in Connection with Scattering Theor…
We consider Jacobi matrices and Schrodinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then…
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on…
We show that for a Jacobi operator with coefficients whose (j+1)'th moments are summable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve…
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of…
In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute…
The trace formula for the evolution operator associated with nonlinear stochastic flows with weak additive noise is cast in the path integral formalism. We integrate over the neighborhood of a given saddlepoint exactly by means of a smooth…
We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schr\"odinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the…
We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a…
The diffraction trace formula ({\em Phys. Rev. Lett.} {\bf 73}, 2304 (1994)) and spectral determinant are tested on the open three disk scattering system. The system contains a generic and exponentially growing number of diffraction…
Diffractive systems are quantum-mechanical models with point-like singularities where usual semiclassical approximation breaks down. An overview of recent investigations of such systems is presented. The following examples are considered in…
Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit…
Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…
Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later in 1988, a similar formula was obtained by Neidhardt \cite{NH} in the case of…
In this paper we investigate the spectral and the scattering theory of Schr\"odinger operators acting on perturbed periodic discrete graphs. The perturbations considered are of two types: either a multiplication operator by a short-range or…
We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass-spring system and the perturbation is obtained by modifying one…
This paper studies trace class perturbation of closed linear relations in Hilbert spaces. The concept of trace class perturbation of closed relations is introduced by orthogonal projections. Equivalent characterizations of compact and trace…
Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance…
We study the behaviour of functions of dissipative operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of analytic relatively operator Lipschitz functions. An essential role is played…
Semiclassical trace formulas are examined using phase space path integrals. Our main concern in this paper is the Maslov index of the periodic orbit, which seems not fully understood in previous works. We show that the calculation of the…
The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside…