In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,π^2)$) of the closed bidisc.
We consider a class of compact Toeplitz operators on the Bergman space on the unit disk. The symbols of operators in our class are assumed to have a sufficiently regular power-like behaviour near the boundary of the disk. We compute the asymptotics of singular values of this class of Toeplitz operators. We use this result to obtain the asymptotics of singular values for a class of banded matrices.
We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a conjecture of Valent about the order of a certain class of Jacobi matrices with polynomial coefficients.
In this paper, we consider the Perron theorem over the real Puiseux field. We introduce a recursive method for calculating Perron roots and Perron vectors of positive Puiseux matrices (which satisfy some condition of genericness) by means of combinatorics based on the tropical linear algebra.
We consider Sturm-Liouville equation $y^{\prime\prime}+(λ-q)y=0$ where $q\in L^{1}[0,a]$. We obtain various conditions on the Fourier coefficients of q such that the periodic eigenvalues having the form given by Titchmarsh are asymptotically simple. Under these conditions, we give some asymptotic estimates for the spectral gaps.
We prove uniform resolvent estimates for an abstract operator given by a dissipative perturbation of a self-adjoint operator in the sense of forms. For this we adapt the commutators method of Mourre. We also obtain the limiting absorption principle and uniform estimates for the derivatives of the resolvent. This abstract work is motivated by the Schr{ö}dinger and wave equations on a wave guide with dissipation at the boundary.
This paper contains a re-evaluation of the spectral approach and factorizability for regular matrix polynomials. In addition, solvent theory is extended from the monic and comonic cases to the regular case. The classification of extended solvents (bisolvents) is shown to be equivalent to the classification of all the regular first order right factors for a general matrix polynomial.
In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in space of vector-functions by the Sturm-Liouville equation with m by m matrix potential and the boundary conditions whose scalar case (m=1) are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.
We consider the Schrodinger operator with a constant magnetic field in the exterior of a compact domain on the plane. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We discuss the rate of accumulation of eigenvalues in a fixed cluster.
Let V_0 be a real-valued function on [0,\infty) and V\in L^1([0,R]) for all R>0 so that H(V_0)= -\f{d^2}{dx^2}+V_0 in L^2([0,\infty)) with u(0)=0 boundary conditions has discrete spectrum bounded from below. Let \calM (V_0) be the set of V so that H(V) and H(V_0) have the same spectrum. We prove that \calM(V_0) is connected.
We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application we determine all resonances generated by a saddle point in a fixed disc around the critical energy.
In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators.
It is proved recently by Benamara-Nikolski that a contraction having finite defects and spectrum not filling in the closed unit disc, is similar to a normal operator if and only if it has the so-called linear resolvent growth property. We obtain results of the same type for a wider (than contractions) class of operators.
This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete inverse spectral problems.
In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting.
This paper sets out to study the spectral minimum for operator belonging to the family of random Schrödinger operators of the form $H\_{λ,ω}=-Δ+W\_{\text{per}}+λV\_ω$, where we suppose that $V\_ω$ is of Anderson type and the single site is assumed to be with an indefinite sign. Under some assumptions we prove that there exists $λ\_0>0$ such that for any $λ\in [0,λ\_0]$, the minimum of the spectrum of $H\_{λ,ω}$ is obtained by a given realization of the random variables.
We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previously studied by Froese-Hislop-Perry.