Spectral curves of non-hermitian hamiltonians

Recent analytical and numerical work have shown that the spectrum of the ra ndom non-hermitian Hamiltonian on a ring which models the physics of vortex line pinning in superconductors is one dimensional. In the maximally non-h ermitian limit, we give a simple "one-line" proof of this feature. We then study the spectral curves for various distributions of the random site ener gies. We find that a critical transition occurs when the average of the log arithm of the random site energy squared vanishes. For a large class of pro bability distributions of the site energies, we find that as the randomness increases the energy E-* at which the localization-delocalization transiti on occurs increases, reaches a maximum, and then decreases. The Cauchy dist ribution studied previously in the literature does not have this generic be havior. We determine gamma(c1), the critical value of the randomness at whi ch "wings" first appear in the energy spectrum. For distributions, such as Cauchy, with infinitely long tails, we show that gamma(c1) = 0(+). We deter mine the density of eigenvalues on the wings for any probability distributi on. We show that the localization length on the wings diverge generically a s L(E) similar to 1/\E - E-*\ as E approaches E-*. These results are all ob tained in the maximally non-hermitian limit but for a generic class of prob ability distributions of the random site energies, (C) 1999 Elsevier Scienc e B.V. All rights reserved.