Quantum mechanics with random imaginary scalar potential

Abstract

We study spectral properties of a non-Hermitian Hamiltonian describing a quantum particle propagating in a random imaginary scalar potential. Cast in the form of an effective field theory, we obtain an analytical expression for the ensemble averaged one-particle Green function from which we obtain the density of complex eigenvalues. Based on the connection between non-Hermitian quantum mechanics and the statistical mechanics of polymer chains, we determine the distribution function of a self-interacting polymer in dimensions d > 4.

Extended tail states in an imaginary random potential

“…This transition is usually referred as the mobility edge in 1d problem (which is already surprising [12]). In this paper we consider another example, where the localized(delocalized) eigenfunctions of non-Hermitean Hamiltonian behave in the "inverted", compared to the Hermitean case, way.Two recent papers [10,11] deals with the non-Hermitean quantum mechanical Hamiltonian with an imaginary random potentialThis study was motivated by the observation that the Euclidean evolution operator r| exp[−tH]|0 with H (1) after averaging over δ-correlated disordered potential V coincides with the probability distribution Z(r, t) for the Edwards self-repulsing polymer [13]. Many applications of non-Hermitean operators (like both our examples) come from the statistical physics and naturally deal with the imaginary time(Euclidean) evolution.…”

Optimal Fluctuations and Tail States of Non-Hermitian Operators

“…In fact, the matrix structure ofĤ bares much in common with the Gor'kov or Bogoliubov-de Gennes quasi-particle Hamiltonian of a weakly disordered superconductor [15]. To foster this analogy let us implement the canonical transformation…”

“…Substituted back into the action, these fluctuations are described by the corresponding low-energy action [15] …”

“…Over recent years the statistical properties of stochastic non-Hermitian operators have come under intense scrutiny [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Operators of this kind appear in a number of physical applications ranging from random classical dynamics, and statistical physics, to phase breaking and relaxation in quantum dynamics.…”

“…Beginning with early work on random matrix ensembles [26,27], a variety of techniques have been developed to study the statistical properties of stochastic non-Hermitian operators [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Some of the techniques have been based on the random matrix theory [26,27,1,2,6,7,8,14], while others have involved the development of perturbative schemes, such as the self-consistent Born approximation in the diagrammatic analysis (e.g., Ref.…”