On Hubbard-Stratonovich Transformations over Hyperbolic Domains

We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with "chiral" symmetry. Apart from this we outline a way of deriving the nonlinear $\sigma$-model from the gauge-invariant Wegner $k-$orbital model avoiding the use of the HS transformations.Comment: More accurate proofs are given; a few misprints are corrected; a misleading reference and a footnote in the end of section 2.2 are remove

Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality

We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the non-orthogonality of resonance wavefunctions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems, and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum.Comment: 4 pages, 1 figure (published version with minor changes

A conjecture on Hubbard-Stratonovich transformations for the Pruisken-Sch\"afer parameterisations of real hyperbolic domains

Rigorous justification of the Hubbard-Stratonovich transformation for the Pruisken-Sch\"afer type of parameterisations of real hyperbolic O(m,n)-invariant domains remains a challenging problem. We show that a naive choice of the volume element invalidates the transformation, and put forward a conjecture about the correct form which ensures the desired structure. The conjecture is supported by complete analytic solution of the problem for groups O(1,1) and O(2,1), and by a method combining analytical calculations with a simple numerical evaluation of a two-dimensional integral in the case of the group O(2,2).Comment: Published versio

The decay of photoexcited quantum systems: a description within the statistical scattering model

The decay of photoexcited quantum systems (examples are photodissociation of molecules and autoionization of atoms) can be viewed as a half-collision process (an incoming photon excites the system which subsequently decays by dissociation or autoionization). For this reason, the standard statistical approach to quantum scattering, originally developed to describe nuclear compound reactions, is not directly applicable. Using an alternative approach, correlations and fluctuations of observables characterizing this process were first derived in [Fyodorov YV and Alhassid Y 1998 Phys. Rev. A 58, R3375]. Here we show how the results cited above, and more recent results incorporating direct decay processes, can be obtained from the standard statistical scattering approach by introducing one additional channel.Comment: 7 pages, 2 figure

Correlation functions of impedance and scattering matrix elements in chaotic absorbing cavities

Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.Comment: 10 pages, 2 figures (misprints corrected and references updated in ver.2); to appear in Acta Phys. Pol. A (Proceedings of the 2nd Workshop on Quantum Chaos and Localization Phenomena, May 19-22, 2005, Warsaw

On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U(N)}$ according to the Haar measure.Comment: 10 pages, 1 figur

Random Energy Model with complex replica number, complex temperatures and classification of the string's phases

The results by E. Gardner and B.Derrida have been enlarged for the complex temperatures and complex numbers of replicas. The phase structure is found. There is a connection with string models and their phase structure is analyzed from the REM's point of view.Comment: 11 pages,revte

Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems

We investigate systematically sample-to sample fluctuations of the probability $\tau$ of no return into a given entrance channel for wave scattering from disordered systems. For zero-dimensional ("quantum chaotic") and quasi one-dimensional systems with broken time-reversal invariance we derive explicit formulas for the distribution of $\tau$, and investigate particular cases. Finally, relating $\tau$ to violation of S-matrix unitarity induced by internal dissipation, we use the same quantity to identify the Anderson delocalisation transition as the phenomenon of spontaneous breakdown of S-matrix unitarity.Comment: This is the published version, with a few modifications added to the last par