Spectra of random matrices close to unitary and scattering theory for discrete-time systems

Abstract

We analyze statistical properties of complex eigenvalues of random matrices close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. Deviation from unitarity are characterized by rank M and eigenvalues Ti, i = 1, ..., M of the matrixT =1 − †Â . For the case M = 1 we solve the problem completely by deriving the joint probability density of eigenvalues and calculating all n− point correlation functions. For a general case we present the correlation function of secular determinants. PACS numbers: 03.65 Nk, 05.45 MtThe theory of wave scattering can be looked at as an integral part of the general theory of linear dynamic open systems in terms of the input-output approach. These ideas and relations were developed in system theory and engeneering mathematics many years ago, see papers [1-3] and references therein. Unfortunately, that development went almost unnoticed by the majority of physicists working in the theory of chaotic quantum scattering and related phenomena, see [4,5] and references therein. For this reason I feel it could be useful to recall some basic facts of the input-output approach in such a context.An Open Linear System is characterized by three Hilbert spaces: the space E 0 of internal states Ψ ∈ E 0 and two spaces E ± of incoming (-) and outgoing (+) signals or waves also called input and output spaces, made of vectors φ ± ∈ E ± . Acting in these three spaces are four operators, or matrices: a) the so-called fundamental operator which maps any vector from internal space E 0 onto some vector from the same space E 0 b) two op-eratorsŴ 1,2 , withŴ 1 mapping incoming states onto an internal state andŴ 2 mapping internal states onto outgoing states and c) an operatorŜ 0 acting from E − to E + .We will be interested in describing the dynamics Ψ(t) of an internal state with time t provided we know the state at initial instant t = 0 and the system is subject to a given input signal φ − (t). In what follows we consider only the case of the so-called stationary (or time-invariant) systems when the operators are assumed to be time-independent. Let us begin with the case of continous-time description. The requirements of linearity, causality and stationarity lead to a system of two dynamical equations:Interpretation of these equations depends on the nature of state vector Ψ as well as of vectors φ ± and is different in different applications. In the context of quantum mechanics one relates the scalar product Ψ † Ψ with the probability to find a particle inside the "inner" region at time t, whereas φ † ± φ ± stays for probability currents flowing in and out of the region of internal states (the number of particles coming or leaving the inner domain per unit time). The condition of particle conservation then reads as:It is easy to verify that Eq.(2) is compatible with the dynamics Eq.(1) only provided the operators satisfy the following relations:which shows, in particular, that can be written as A = ...

Random matrices close to Hermitian or unitary: overview of methods and results

“…Similar effects were discussed in relation to the unimolecular decay in N O 2 molecules [21]. In section 4 we discuss this issue for the regime of weak non-Hermiticity.Most of our own results discussed in this review were reported earlier in the form of short communications [36,51,52,57,61,62,63] and in an unpublished paper by one of the authors [64]. Some (but by far not all) details of the underlying derivations are elucidated in the present text, as far as space restrictions allow.…”

“…(96) is that it allows for calculation of all n−point correlation functions for arbitary N, n, M with help of the method of orthogonal polynomials. Again, the particular case M = 1 [62] is quite instructive and can be recommended to follow first for understanding of the general formulae outlined below.…”

“…(7). We present the final result, which has the form of a Hankel determinant, in terms of eigenvalues G i = 1 − T i ofĜ:The particular case n = 1 of the above formula was presented in [62]. In fact, the expression Eq.…”

“…A particular case which can be solved relatively easy is the case of a rank-one deviation from Hermiticity: M = 1, corresponding to a system with only one open channel. This example is quite instructive and the exposition below follows [62]. Evaluating the integrals Eq.…”

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

“…. , 1) with |a| < 1 was calculated by Fyodorov [6] (see also [7]), up to a proportionality. The starting point was the formula for the joint eigenvalue distribution…”