Perturbation of operators and approximation of spectrum

Abstract

Let A (x) be a norm continuous family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)(n) be the orthogonal compressions of A (x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)(n). We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)(n). The known results, for a bounded self-adjoint operator, are translated into the case of a norm continuous family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of norm continuous perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here

Approximations of Strongly Continuous Families of Unbounded Self-Adjoint Operators

“…Both papers provide extensive references to additional literature in the field. We also mention [9], where analysis similar to ours is performed for bounded operators. We note that spectral pollution (the appearance of spurious eigenvalues within gaps in the essential spectrum when approximating) has attracted significant attention [4,10,11].…”

A note on discrete Borg-type theorems

“…The proof techniques are not much different, but we have to take care of the convergence issues of the matrix entries. Theorem 4.1 of [8] can be modified in the following way.…”

A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

“…Since there are no eigenvalues when = 0, by general results of perturbation theory, eigenvalues of can appear in a gap only by emerging from one of its end points as is varied. Likewise, eigenvalues can disappear from the gap only by converging to an end point [14][15][16][17][18]. We call 0 a coupling constant threshold of the family of the operators at the gap end points and if there exists an eigenvalue branch ( ) of such that ( ) ↓ or ( ) ↑ , as either ↑ 0 or ↓ 0 , respectively, or both.…”