On the spectral behaviour of a non-self-adjoint operator with complex potential
type: Item
instance of: scholarly article
title: On the spectral behaviour of a non-self-adjoint operator with complex potential
zbMATH Open document ID: 1181.47026
author: Carmen Martínez Adame
published in: Mathematical Physics, Analysis and Geometry
publication date: 2009-05-22T00:00:00Z
review text: A sequence of complex numbers \(\{v_n\}\), \(n\in\mathbb{Z}\), which takes values in a finite set \(M\) is called pseudo-ergodic if, for any positive integer \(N\) and an arbitrary set \(w_1,\dots,w_N\) from \(M\), there is \(q\in\mathbb{Z}\) such that \(v_{q+j}=w_j\), \(j=1,2,\dots,N\). The author studies the properties of the spectrum of a nonselfadjoint Jacobi operator \(J\) on \(\ell^2(\mathbb{Z})\) with constant off-diagonals \(\alpha,\beta\in\mathbb{R}\) and a pseudo-ergodic main diagonal \(v_n\) with the 4-point set \(M=\{\pm\gamma\pm i\delta\}\), \(\gamma,\delta>0\). By using the second order numerical range, the author obtains tight bounds on the spectrum and finds the bounds for the size of possible holes inside the spectrum.
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47B80
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 5556070
zbMATH Keywords: complex potential
zbMATH Keywords: pseudo-ergodic operators
zbMATH Keywords: numerical range
zbMATH Keywords: spectrum
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: s11040-007-9022-7
OpenAlex ID: W2079191368
cites work: Spectral properties of random non-self-adjoint matrices and operators
cites work: Spectral theory of pseudo-ergodic operators.
cites work: Spectral Bounds Using Higher-Order Numerical Ranges
cites work: \(L^ 1\)-properties of intrinsic Schrödinger semigroups
cites work: Eigenvalue curves of asymmetric tridiagonal random matrices
cites work: Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model
cites work: Q3416487
DOI: 10.1007/S11040-007-9022-7
description: scientific article; zbMATH DE number 5556070