Abstract:
The recently developed quantum surface of section method is applied to a search for extremely high-lying energy levels in a simple but generic Hamiltonian system between integrability and chaos, namely the semiseparable 2-dim oscillator. Using the stretch of 13,445 consecutive levels with the sequential number around $1.8\cdot 10^7$ (eighteen million) we have clearly demonstrated the validity of the semiclassical Berry-Robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power-law level repulsion which is globally very well described by the Brody distribution.

Abstract:
The paper presents exact surface of section reduction of quantum mechanics. The main theoretical result is a decomposition of the energy-dependent propagator G(E) = (E - H)^(-1) in terms of the propagators which (also or exclusively) act in Hilbert space of complex-valued functions over the configurational surface of section, which has one dimension less than the original configuration space. These energy-dependent quantum propagators from and/or onto the configurational surface of section can be explicitly constructed as the solutions of the first order nonlinear Riccati-like initial value problems.

Abstract:
A new method for exact quantization of general bound Hamiltonian systems is presented. It is the quantum analogue of the classical Poincare Surface Of Section (SOS) reduction of classical dynamics. The quantum Poincare mapping is shown to be the product of the two generalized (non-unitary but compact) on-shell scattering operators of the two scattering Hamiltonians which are obtained from the original bound one by cutting the f-dim configuration space (CS) along (f-1)-dim configurational SOS and attaching the flat quasi-one-dimensional waveguides instead. The quantum Poincare mapping has fixed points at the eigenenergies of the original bound Hamiltonian. The energy dependent quantum propagator (E - H)^(-1) can be decomposed in terms of the four energy dependent propagators which propagate from and/or to CS to and/or from configurational SOS (which may generally be composed of many disconnected parts). I show that in the semiclassical limit (hbar ---> 0) the quantum Poincare mapping converges to the Bogomolny's propagator and explain how the higher order semiclassical corrections can be obtained systematically.

Abstract:
Local parametric statistics of zeros of Husimi representations of quantum eigenstates are introduced. It is conjectured that for a classically fully chaotic systems one should use the model of parametric statistics of complex roots of Gaussian random polynomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are predicted to obey a universal (non-Maxwellian) distribution ${d P(v)}/{dv^2} = 2/(\pi\sigma^2)(1 + |v|^2/\sigma^2)^{-3},$ where $\sigma^2$ is the mean square velocity. The conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom. Dynamical formulation of the ``zero-flow'' in terms of an integrable many-body dynamical system is given as well.

Abstract:
k-point correlations of complex zeros for Gaussian ensembles of Random Polynomials of order N with Real Coefficients (GRPRC) are calculated exactly, following an approach of Hannay for the case of Gaussian Random Polynomials with Complex Coefficients (GRPCC). It is shown that in the thermodynamic limit $N \to \infty$ of Gaussian random holomorphic functions all the statistics converge to the their GRPCC counterparts as one moves off the real axis, while close to the real axis the two cases are essentially different. Special emphasis is given to 1 and 2 point correlation functions in various regimes.

Abstract:
Numerical calculation and analysis of extremely high-lying energy spectra, containing thousands of levels with sequential quantum number up to 62,000 per symmetry class, of a generic chaotic 3D quantum billiard is reported. The shape of the billiard is given by a simple and smooth de formation of a unit sphere which gives rise to (almost) fully chaotic classical dynamics. We present an analysis of (i) quantum length spectrum whose smooth part agrees with the 3D Weyl formula and whose oscillatory part is peaked around the periods of classical periodic orbits, (ii) nearest neighbor level spacing distribution and (iii) number variance. Although the chaotic classical dynamics quickly and uniformly explores almost entire energy shell, while the measure of the regular part of phase space is insignificantly small, we find small but significant deviations from GOE statistics which are explained in terms of localization of eigenfunctions onto lower dimensional classically invariant manifolds.

Abstract:
This is the first survey of highly excited eigenstates of a chaotic 3D billiard. We introduce a strongly chaotic 3D billiard with a smooth boundary and we manage to calculate accurate eigenstates with sequential number (of a 48-fold desymmetrized billiard) about 45,000. Besides the brute-force calculation of 3D wavefunctions we propose and illustrate another two representations of eigenstates of quantum 3D billiards: (i) normal derivative of a wavefunction over the boundary surface, and (ii) ray - angular momentum representation. The majority of eigenstates is found to be more or less uniformly extended over the entire energy surface, as expected, but there is also a fraction of strongly localized - scarred eigenstates which are localized either (i) on to classical periodic orbits or (ii) on to planes which carry (2+2)-dim classically invariant manifolds, although the classical dynamics is strongly chaotic and non-diffusive.

Abstract:
The spectral theorem is proven for the quantum dynamics of quadratic open systems of n fermions described by the Lindblad equation. Invariant eigenspaces of the many-body Liouvillean dynamics and their largest Jordan blocks are explicitly constructed for all eigenvalues. For eigenvalue zero we describe an algebraic procedure for constructing (possibly higher dimensional) spaces of (degenerate) non-equilibrium steady states.

Abstract:
The unitary representation of exact quantum Poincare mapping is constructed. It is equivalent to the compact representation in a sense that it yields equivalent quantization condition with important advantage over the compact version: since it preserves the probability it can be literally interpreted as the quantum Poincare mapping which generates quantum time evolution at fixed energy between two successive crossings with surface of section (SOS). SOS coherent state representation (SOS Husimi distribution) of arbitrary (either stationary or evolving) quantum SOS state (vector from the Hilbert space over the configurational SOS) is introduced. Dynamical properties of SOS states can be quantitatively studied in terms of the so called localization areas which are defined through information entropies of their SOS coherent state representations. In the second part of the paper I report on results of extensive numerical application of quantum SOS method in a generic but simple 2-dim Hamiltonian system, namely semiseparable oscillator. I have calculated the stretch of 13500 consecutive eigenstates with the largest sequential quantum number around 18 million and obtained the following results: (i) the validity of the semiclassical Berry-Robnik formula for level spacing statistics was confirmed and using the concept of localization area the states were quantitatively classified as regular or chaotic, (ii) the classical and quantum Poincare evolution were performed and compared, and expected agreement was found, (iii) I studied few examples of wavefunctions and particularly, SOS coherent state representation of regular and chaotic eigenstates and analyzed statistical properties of their zeros which were shown on the chaotic component of 2-dim SOS to be uniformly distributed with the cubic repulsion between nearest neighbours.

Abstract:
The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n x 4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field.