Spectral Theory of Pseudo-Ergodic Operators

Abstract

We define a class of pseudo-ergodic non-self-adjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonself-adjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.

Regular Spacings of Complex Eigenvalues in the One-Dimensional Non-Hermitian Anderson Model

“…A very simple calculation shows then that the spectrum of H g ∞ would typically contain (with probability 1) a two-dimensional subset of the complex plain. (A much more detailed description of the spectrum of the limiting operator can be found in [3].) However, numerical experiments reproduce pictures like that in Fig.1 with remarkable stability also for large values of n (in [5,6] n = 1000).…”

Spectral Bounds Using Higher-Order Numerical Ranges

“…On the other hand, the present author studied the same model, defined directly as a bounded linear operator acting on l 2 (Z D ) for any D, and found that the spectrum consists of a bounded region in the complex plane; this is described in some detail when D = 1 in [9,10]. Nevertheless, a number of spectral questions still needed to be clarified.…”

“…Also, Spec(A) ⊂ E and Spec(A) → E as M → ∞. If γ > 0 and A ∞ is pseudo-ergodic in the sense of [10], then…”

“…From this point of view, the results of Goldsheid and Khoruzhenko are sharper, while the D model provides generic bounds on the asymptotic spectra of HN models for which the sharper results have not yet been computed, or are beyond the methods of those papers because, for example, αβ < 0. In a separate paper [22], Martínez studies the D model, and obtains considerable improvements on the bounds given in [9,10].…”

Spectral Properties of Non-Unitary Band Matrices

“…However, the spectral theory of non-self-adjoint operators has been the object of many works, in various setups of regimes, as can be seen from the papers [6][7][8][10][11][12]16,31,32,34] and references therein. In particular, several analyses of non-self-adjoint operators focus on tri-diagonal operators, when expressed in a certain basis; see [7,8,10,11]. Since Jacobi matrices provide generic models of self-adjoint operators, it is quite natural to deal with non-self-adjoint tri-diagonal matrices which are deformations of Jacobi matrices.…”