Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwords

Abstract

We prove criteria, purely based on finite subwords of the potential, for spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D: By employing localized lower norms, we strongly generalize known results from the self-adjoint case to much more general and non-normal situations, including various configurations of Hamiltonians and further non-self-adjoint models with aperiodic or pseudoergodic potentials, even models with multiple varying diagonals and entries in a Banach space.

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

“…C C is the mapping given by ".A/ WD Spec A for A 2 ‰ , the computational problem ¹"; ‰ ; C C ; ƒº has arithmetic SCI, in the sense of [2,25], equal to one; more precisely, since Spec A y … n fin .A/, for each n 2 N and A 2 ‰ , this computational problem is in the class … A 1 , as defined in [2,25]. To make this concrete (see the related discussions in [14, Section 4.3] and [44]), consider the Feinberg-Zee case that † ˛D ¹ 1; 1º, † ˇD ¹0º, and † D ¹1º, and suppose A, taking the form (1.15), is a random Feinberg-Zee matrix, with ˇj D 0 and j D 1, for j 2 Z, and with the entries ˛j 2 ¹ 1; 1º i.i.d. random variables, with p WD Pr.˛j D 1/ 2 .0; 1/.…”

“…has cardinality jà . / n j that is finite but grows exponentially with n. There are interesting potential applications, notably the Fibonacci Hamiltonian and nonself-adjoint variants [27,28,44], where jà . / n j grows only linearly with n, allowing exploration of spectral properties with much larger values of n. At the other extreme, where à n (and the corresponding and 1 method sets) are uncountable and so must be approximated as in Theorems 6.3 and 7.1, one might seek to use or adapt our methods to study the spectra of (bounded) integral operators A on L 2 .R/, for example those that arise in problems of wave scattering by unbounded rough surfaces (e.g., [87]).…”