Related papers: Inverse Problems for Jacobi Operators with Mixed S…
We study the inverse spectral problem for periodic Schr\"odinger opera\-tors of kind $- \frac{1}{2} \hbar^2 \Delta_x + V(x)$ on the flat torus $\Bbb T^n := (\Bbb R / 2 \pi \Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. We show…
Boundary value problems on hedgehog-type graphs for Sturm-Liouville differential operators with general matching conditions are studied. We investigate inverse spectral problems of recovering the coefficients of the differential equation…
The inverse eigenvalue problem for real symmetric matrices of the form 0 0 0 . 0 0 * 0 0 0 . 0 * * 0 0 0 . * * 0 . . . . . . . 0 0 * . 0 0 0 0 * * . 0 0 0 * * 0 . 0 0 0 is solved. The solution is shown to be unique. The problem is also…
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2x2 canonical systems). We prove a number of Szeg\H{o}-type theorems for…
In this paper we approximate a Schr\"odinger operator on $L^2(\R)$ by Jacobi operators on $\ell^2(\Z)$ to provide new proofs of sharp Lieb-Thirring inequalities for the powers $\gamma=1/2$ and $\gamma=3/2$. To this end we first investigate…
We explore to what extent the relation between the absolute continuous spectrum and non-existence of subordinate generalized eigenvectors, known for scalar Jacobi operators, can be formulated also for block Jacobi operators with…
We find asymptotics of entries of Jacobi matrices with lacunary spectral data under some additional growth conditions. We also prove the inverse results. In addition, we study connections between Jacobi matrices, canonical systems and de…
We describe the spectral properties of the Jacobi operator $(Hy)_n= a_{n-1} y_{n-1}+a_{n}y_{n+1}+b_ny_n,$ $n\in\Z,$ with $a_n=a_n^0+ u_n,$ $b_n= b_n^0+ v_n,$ where sequences $a_n^0>0,$ $b_n^0\in\R$ are periodic with period $q$, and…
We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous…
We provide an explicit spectral representation for several weighted Hankel matrices by means of the so called commutator method. These weighted Hankel matrices are found in the commutant of Jacobi matrices associated with orthogonal…
We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential…
A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on…
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…
We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In…
The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs.…
In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form \[P( (-\Delta_g)^s)u := \sum_{i=1}^M \alpha_i(-\Delta_{g_i})^{s_i}u\] for $s=(s_1,\dots,s_M)$,…
We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain…
In this paper inverse problems for Dirac operator with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are provided, which are generalizations of the well-known Weyl…
Variable order differential equations with non-integrable singularities are considered on spatial networks. Properties of the spectrum are established, and the solution of the inverse spectral problem is obtained.