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In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation…
In this paper, a family of random Jacobi matrices, with off-diagonal terms that exhibit power-law growth, is studied. Since the growth of the randomness is slower than that of these terms, it is possible to use methods applied in the study…
This work studies the interplay between Green functions, the index of determinacy of spectral measures and interior finite rank perturbations of Jacobi operators. The index of determinacy quantifies the stability of uniqueness of solutions…
Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators…
Jacobi operators appear as kinetic operators of several classes of noncommutative field theories (NCFT) considered recently. This paper deals with the case of bounded Jacobi operators. A set of tools mainly issued from operator and spectral…
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the…
The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the…
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring…
We consider the dynamic problems for the discrete systems with discrete time associated with finite and semi-infinite Jacobi matrices. The result of the paper is a procedure of association of special Hilbert spaces of functions, namely de…
We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the…
We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown that given a set of $n$ not necessarily distinct non-real numbers in the open upper…
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary conditions leads to explicit formulas for the averaged spectral measure which can…
We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the…
In this paper we consider eigenvalues asymptotics of the energy operator in the one of the most interesting models of quantum physics, describing an interaction between two-level system and harmonic oscillator. The energy operator of this…
We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on…
The Jacobi matrices with bounded elements whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum are considered. A new type of spectral data, which are analogous for…
We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal…
We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of…
The research on spectral inequalities for discrete Schrodinger Operators has proved fruitful in the last decade. Indeed, several authors analysed the operator's canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we…
We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic ("double root") situation. For the model with "non-smooth" matrix entries we obtain the asymptotics of generalized eigenvectors and analyze…