Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators
نویسندگان
چکیده
We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator − d dx +V (x) on L2(0,∞) with V ∈ L2(0,∞) and u(0) = 0 boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.
منابع مشابه
A spectral equivalence for Jacobi matrices
We use the classical results of Baxter and Gollinski-Ibragimov to prove a new spectral equivalence for Jacobi matrices on l(N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences, and find necessary and sufficient conditions on the spectral measure such that ∑ ∞ k=n bk and ∑ ∞ k=n (a k − 1) lie in l 1 ∩ l .
متن کاملComparative study on solving fractional differential equations via shifted Jacobi collocation method
In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...
متن کاملA Strong Szegő Theorem for Jacobi Matrices
We use a classical result of Gollinski and Ibragimov to prove an analog of the strong Szegő theorem for Jacobi matrices on l2(N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that ∑ ∞ k=n bk and ∑ ∞ k=n(a 2 k − 1) lie in l2 1, the linearly-weighted l2 space.
متن کاملA ug 2 00 9 1 – D Schrödinger operators with local interactions on a discrete set
Spectral properties of 1-D Schrödinger operators HX,α := − d 2 dx2 + ∑ xn∈X αnδ(x − xn) with local point interactions on a discrete set X = {xn}n=1 are well studied when d∗ := infn,k∈N |xn − xk| > 0. Our paper is devoted to the case d∗ = 0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl funct...
متن کاملGeneralized Continuous Frames for Operators
In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ wi...
متن کامل