The Essential Spectrum of Schrödinger, Jacobi, and Cmv Operators Yoram Last and Barry Simon
نویسنده
چکیده
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 spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomials with finite essential spectra.
منابع مشابه
The Essential Spectrum of Schrödinger, Jacobi, and Cmv Operators
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 spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomial...
متن کاملFine Structure of the Zeros of Orthogonal Polynomials, Iv. a Priori Bounds and Clock Behavior Yoram Last and Barry Simon
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.
متن کاملEssential Closures and Ac Spectra for Reflectionless Cmv, Jacobi, and Schrödinger Operators Revisited
We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application of the notion of the essential closure of subsets of R we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set E of positiv...
متن کاملEigenfunctions, Transfer Matrices, and Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators
on L(R ) (and its half-line problem, H+, on L2(0,∞) with u(0) = 0 boundary conditions). We will focus on a new approach to the absolutely continuous spectrum σac(h) and, more generally, Σac(h), the essential support of the a.c. part of the spectral measures. What is new in our approach is that it relies on estimates on the transfer matrix, that is, the 2× 2 matrix TE(n,m) which takes (u(m+1) u(...
متن کاملEigenvalue Bounds in the Gaps of Schrödinger Operators and Jacobi Matrices
We consider C = A+B where A is selfadjoint with a gap (a, b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV ) bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.
متن کامل