Minimal representations of unitary operators and orthogonal polynomials on the unit circle

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.Comment: 31 page

Spectral Stability of Unitary Network Models

We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory

On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we focus on the "hopping sign model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random $\pm 1$'s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and $p$-norm \eps-pseudospectra (\eps>0, $p\in [1,\infty]$) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum $\Sigma$. We also propose a sequence of inclusion sets for $\Sigma$ which we show is convergent to $\Sigma$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $n\times n$ matrices. We propose similar convergent approximations for the 2-norm \eps-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below

Effective inverse spectral problem for rational Lax matrices and applications

We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights nu and 1-nu respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of ``elementary twists'', we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite--band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance of formulae obtained to the study of the large degree asymptotics for these polynomials is indicated.Comment: 33 pages. Version 2 with added references suggested by the refere

Finite sections of the Fibonacci Hamiltonian

We study finite but growing principal square submatrices $A_n$ of the one- or two-sided infinite Fibonacci Hamiltonian $A$. Our results show that such a sequence $(A_n)$, no matter how the points of truncation are chosen, is always stable -- implying that $A_n$ is invertible for sufficiently large $n$ and $A_n^{-1}\to A^{-1}$ pointwise