Related papers: Spectral estimates for 2D periodic Jacobi operator…
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger operator (FSO). In the problem, in order to get reliable gaps distribution…
We consider Laplacians on periodic metric graphs with unit-length edges. The spectrum of these operators consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite…
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…
We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\equiv 1$, $b_n =-C n^{-\beta}$ ($0<\beta< \frac23)$, one has $d\mu(x)= w(x) dx$ on $(-2,2)$, and near…
A practical solution for the mathematical problem of functional calculus with Laplace-Beltrami operator on surfaces with axial symmetry is found. A quantitative analysis of the spectrum is presented.
We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in…
An algorithm is presented which generates pairs of oscillatory random time series which have identical periodograms but differ in the number of oscillations. This result indicate the intrinsic limitations of spectral methods when it comes…
Two method for computation of the spectra of certain infinite graphs are suggested. The first one can be viewed as a reversed Gram--Schmidt orthogonalization procedure. It relies heavily on the spectral theory of Jacobi matrices. The second…
We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive…
We study bounds on eigenvalue gaps for finite quotients of periodic Jacobi matrices on trees. We prove an Alon-Boppana type bound for the spectral gap and a comparison result for other eigenvalue gaps.
In this paper we establish a Lebeau-Robbiano spectral inequality for a degenerated one dimensional elliptic operator and show how it can be used to impulse control and finite time stabilization for a degenerated parabolic equation.…
We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide…
We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows…
We consider semi-infinite Jacobi matrices with discrete spectrum. We prove that the Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and norming constants corresponding to the first spectrum. We…
In this paper, we study the numerical range of Jacobi operators and it is shown that under certain conditions, the boundary of the numerical range of these operators can be non-round only at the points where it touches the essential…
We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $\Gamma\subset\mathbb Z^d$ of full rank. We describe the class of lattices $\Gamma$ for which the operator may have a spectral gap for arbitrarily small…
We use the classical results of Baxter and Gollinski-Ibragimov to prove a new spectral equivalence for Jacobi matrices on $l^2(\N)$. In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and…
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…
For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end we…
The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We discuss a series of results showing that Schr\"odinger operators can exhibit spectra that are remarkably thin in…