Spectra of Hamiltonians with Generalized Single-Site Dynamical Disorder

Abstract

Starting from the deformed commutation relations \ba a_q(t) \,a_q^†(s) \ - \ q\,a_q^†(s)\,a_q(t) \ = \ \Gam(t-s) {\bf 1} , \quad -1\ \le \ q\ \le\ 1\nn \ea with a covariance $\Gam(t-s)$ and a parameter $q$ varying between $-1$ and $1$, a stochastic process is constructed which continuously deforms the classical Gaussian and classical compound Poisson process. The moments of these distinguished stochastic processes are identified with the Hilbert space vacuum expectation values of products of $\hat{\om}_q (t) = \gam\,\big(\, a_q(t) + a_q^† (t) \,\big)\;+ \; ξ\, a_q^† (t) a_q(t)$ with fixed parameters $q$, $\gam$ and $ξ$. Thereby we can interpolate between dichotomic, random matrix and classical Gaussian and compound Poisson processes. The spectra of Hamiltonians with single-site dynamical disorder are calculated for an exponential covariance (coloured noise) by means of the time convolution generalized master equation formalism (TC-GME) and the