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The parity-time-reversal (PT )-symmetric quantum mechanics (QM) (PTQM) has developed into a noteworthy area of research. However, to date, most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a timedependent PT -symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a timedependent Schrödinger-like equation of motion for PTQM to the familiar timedependent Schrödinger equation in conventional QM. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional QM. Under our proposed framework, we further study in detail the Berry-phase generation in a class of PT -symmetric two-level systems. It is found that a closed path in the parameter space of PTQM is often associated with an open path in a properly mapped problem in conventional QM. In one interesting case, we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional QM despite the existence of a proper mapping between them. [ABSTRACT FROM AUTHOR]

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity-time (PT )-symmetric matrices. To illustrate the main idea, we first study 2 × 2 complex Hermitian matrix ensembles with O(2)-invariant constraints, yielding novel level-spacing statistics such as singular distributions, the half-Gaussian distribution, distributions interpolating between the GOE (Gaussian orthogonal ensemble) distribution and half-Gaussian distributions, as well as the gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2 × 2 PT -symmetric matrix ensembles with real eigenvalues. In particular, PT -symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of PT - symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian unitary ensemble) statistics or the 'truncated- GUE' statistics [ABSTRACT FROM AUTHOR]

Eigenvalue problems for linear differential equations, such as time-independent Schrödinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation and the first, second, and fourth Painlevé transcendents. It is shown here that the differential equations for the first and second Painlevé transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively. [ABSTRACT FROM AUTHOR]