We revisit the spectral statistics of the C
3--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and GUE statistics depending on the symmetry block. Using high--precision Beyn's contour--integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8x10
5 eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE--GUE correspondence and resolve previously observed deviations in long--range spectral correlations. Furthermore, we analyze phase--space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.
We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate the existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits Gaussian Unitary Ensemble (GUE) like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunnelling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.
A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from C(3v) to C3 . Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.
Focusing on the standard map of the kicked rotor, we investigate the effects of classical stickiness (orbits temporarily confined to a region of a chaotic phase space) on an open quantum system. The regions of stickiness that we identify emerge in the strongly chaotic map. By scanning the system's phase space with a leak, we show that stickiness reduces the degree of delocalization of the quantum states and the decay rate of generic initial states. We find an excellent correspondence between the classical dwell time and finite-time Lyapunov exponents with the quantum dwell time and Wehrl entropies of the quantum states. Knowledge of the structure of the classically chaotic trajectories can thus be used to determine the best placement of leaks to enhance or decrease the degree of delocalization of quantum states and to modify the quantum dynamics of open systems.
We consider the geometry of quantum states associated with random matrix Hamiltonians belonging to ensembles that exhibit an integrable-to-chaotic transition in terms of the nearest-neighbour energy level spacing distribution, focusing on the
β-Gaussian ensembles with generic Dyson index. For the tridiagonal Gaussian
β-ensemble, which shows chaos-to-integrability transition with varying Dyson index, we first calculate an analytical expression for the ensemble-averaged fidelity susceptibility for a two-by-two matrix representation of the Hamiltonian for generic values of the Dyson index, and show that it diverges as the total Hamiltonian, which is the sum of a diagonal matrix with independent elements and a tridiagonal
β-matrix, goes over to the integrable phase. Next, for large-dimensional matrices, we numerically compute the fidelity susceptibility and the quantum metric tensor, respectively, for a one-parameter and a two-parameter class of Hamiltonians that we construct from the
β-ensemble, and find scaling relations of these quantities for chaotic and integrable phases. We also consider variations of the
β-ensemble, such as the one that preserves the rotational invariance of the ensemble, and compute the relevant ensemble-averaged fidelity susceptibility to show that it has similar features as the tridiagonal ensemble in the integrable as well as the chaotic phase, thereby establishing the universality of these properties. Finally, the presence of non-vanishing non-diagonal elements, which arises due to the rotational non-invariance of the
β-ensembles, is a specific feature of the corresponding metric tensor, and we use this to illuminate the difference between the quantum state space geometry of these ensembles and the rotationally invariant ones, such as the classical Gaussian ensembles.
Abstract According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.