Related papers: The Inverse Spectral Problem for Jacobi-Type Penci…
We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of…
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…
Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…
Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.
In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schr\"odinger operators, Laplacians on…
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…
We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…
This paper investigates the spectral properties of Jacobi matrices with limit-periodic coefficients. We show that for a residual set of such matrices, the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi…
We introduce concepts of essential numerical range for the linear operator pencil $\lambda\mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor…
For an arbitrary Hermitian period-$T$ Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, $S$, of the spectral…
For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV…
We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are…
We solve the inverse spectral problem for rotationally symmetric manifolds, which include the class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the…
The present paper deals with the spectral and the oscillation properties of a linear pencil $A-\lambda B$. Here $A$ and $B$ are linear operators generated by the differential expressions $(py")"$ and $-y"+ cry$, respectively. In particular,…
To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the…
In the previous paper, we reviewed the Rosenhain's paper to the Jacobi's inversion problem for the genus two hyperelliptic integral. In this paper, we review the G\"{o}pel's paper to the Jacobi's inversion problem for the genus two…
We construct noncomplete orthogonal systems on the ray $[0,\infty)$ that look like Jacobi polynomials $P_n(x)$ after a shift of degree $n\mapsto n+a$, where $a$ is a real constant. These systems are solutions of some exotic Sturm-Liouville…
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…