There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").

The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hörmander’s commutator condition for partial differential equations, ε-pseudoeigenvectors of such matrices for exponentially small values of ε exist in the form of localized wave packets. The symbol need not be smooth with respect to x, just differentiable at a point (or less).

The timelike boundary Liouville (TBL) conformal field theory consisting of a negative norm boson with an exponential boundary interaction is considered. TBL and its close cousin, a positive norm boson with a non-hermitian boundary interaction, arise in the description of the c = 1 accumulation point of c < 1 minimal models, as the worldsheet description of open string tachyon condensation in string theory and in scaling limits of superconductors with line defects. Bulk correlators are shown to be exactly soluble. In contrast, due to OPE singularities near the boundary interaction, the computation of boundary correlators is a challenging problem which we address but do not fully solve. Analytic continuation from the known correlators of spatial boundary Liouville to TBL encounters an infinite accumulation of poles and zeros. A particular contour prescription is proposed which cancels the poles against the zeros in the boundary correlator d(ω) of two operators of weight ω 2 and yields a finite result. A general relation is proposed between two-point CFT correlators and stringy Bogolubov coefficients, according to which the magnitude of d(ω) determines the rate of open string pair creation during tachyon

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes to the infinite volume limit.

E l e c t r o n

This report is concerned with the union sp Tn (a) of all possible spectra that may emerge when perturbing a large n n Toeplitz band matrix Tn (a) in the (j; k) site by a number randomly chosen from some set The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of sp Tn (a) as n ! 1.

This paper explores the relationship between the spectra of perturbed innite banded Laurent matrices L(a)+K and their approximations by perturbed circulant matrices Cn (a) +PnKPn for large n. The entries K jk of the perturbation matrices assume values in prescribed sets

This paper is concerned with the change of the spectra of innite Toeplitz and Laurent matrices under perturbations in a prescribed nite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not aected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site. 1 Introduction and main results Given a complex-valued continuous function a on the complex unit circle T, a 2 C, we consider the matrices T (a) = (a j k ) 1 j;k=1 and L(a) = (a j k ) 1 j;k=1 ; where a k = 1 2 2 Z 0 a(e i ) e ik d; k 2 Z: The matrices T (a) and L(a) are referred to as the Toeplitz matrix and the Laurent matrix with the symbol a, respectively. These matrices induce bounded operators on l 2 (N) and l 2 (Z), respectively. 1 Email: aboet...

This report investigates the possible spectra of large, nite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left mm block. The main result shows that the asymptotic spectrum of such a matrix is not aected by these perturbations, provided they have suciently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rankone perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit. 1

We describe how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what we call its higher order numerical ranges. We prove some of their basic properties and describe explain how to compute them. We finally use them to obtain new spectral insights into the non-selfadjoint Anderson model in one and two space dimensions.