Fabio Masillo
We study some aspects of the Quantum Brachistochrone Problem. Physical realizability of the faster pseudo Hermitian version of the problem is also discussed. This analysis, applied to simple quantum gates, supports an informational interpretation of the problem that is quasi Hermitian invariant.
http://arxiv.org/abs/1105.3332
Quantum Physics (quant-ph)
Alexandre Eremenko, Andrei Gabrielov
We describe a parametrization of the real spectral locus of the two-parametric family of PT-symmetric quartic oscillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey, Dunning, Tateo and Shin.
http://arxiv.org/abs/1105.2742
Mathematical Physics (math-ph)
Miloslav Znojil
A toy-model quantum system is proposed. At a given integer \(N\) it is defined by the pair of \(N\) by \(N\) real matrices \((H,\Theta)\) of which the first item \(H\) specifies an elementary, diagonalizable non-Hermitian Hamiltonian \(H \neq H^\dagger\) with the real and explicit spectrum given by the zeros of the \(N-\)th Chebyshev polynomial of the first kind. The second item \(\Theta\neq I\) must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation \(H^\dagger \Theta= \Theta\,H\). The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the \(N-\)plet of optional parameters, \(\Theta=\Theta(\vec{\kappa})>0\) which must be (and are being) selected as lying in the positivity domain of the metric, \(\vec{\kappa} \in {\cal D}^{(physical)}\).
http://arxiv.org/abs/1105.1863
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)
Gernot Akemann
We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart-Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian two-matrix models, where the complex eigenvalues of the product of the two independent rectangular matrices are sought, with the matrix elements of both matrices being either real, complex or quaternion real. We also present the more general case depending on a non-Hermiticity parameter, that allows us to interpolate between the corresponding three Hermitian Wishart ensembles with real eigenvalues and the maximally non-Hermitian case. All three symmetry classes are explicitly solved for finite matrix size NxM for all complex eigenvalue correlations functions (and real or mixed correlations for real matrix elements). These are given in terms of the corresponding kernels built from orthogonal or skew-orthogonal Laguerre polynomials in the complex plane. We then present the corresponding three Bessel kernels in the complex plane in the microscopic large-N scaling limit at the origin, both at weak and strong non-Hermiticity with M-N greater or equal to 0 fixed.
http://arxiv.org/abs/1104.5203
Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
Alexandre Eremenko, Andrei Gabrielov
We describe the real quasi-exactly solvable spectral locus of the PT-symmetric quartic using the Nevanlinna parametrization.
http://arxiv.org/abs/1104.4980
Mathematical Physics (math-ph)
Carl M. Bender, S. P. Klevansky
A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum mechanics from bosonic systems (systems for which \(T^2=1\)) to fermionic systems (systems for which \(T^2=-1\)). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT-symmetric matrix representations for operator algebras of the form \(\eta^2=0\), \(\bar{\eta}^2=0\), \(\eta\bar{\eta}+\bar {\eta} =\alpha 1\), where \(\bar{eta}=\eta^{PT} =PT \eta T^{-1}P^{-1}\). It is easy to construct matrix representations for the Grassmann algebra (\(\alpha=0\)). However, one can only construct matrix representations for the fermionic operator algebra (\(\alpha \neq 0\)) if \(\alpha= -1\); a matrix representation does not exist for the conventional value \(\alpha=1\).
http://arxiv.org/abs/1104.4156
Subjects: High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Moncy V. John
The demonstration of complex correspondence principle in a recent work is noted to have the drawback that the complex paths in the classical case and complex eigenpaths in the quantum case are very much dissimilar. Also in the de Broglie-Bohm quantum mechanics, considering the harmonic oscillator coherent states, there are marked deviations from classical correspondence. In this Letter, we demonstrate this principle by showing that the complex trajectories of classical harmonic oscillator and the complex quantum trajectories in harmonic oscillator coherent state (in a modified de Broglie-Bohm approach) are identical to each other. Such congruence, which is there even for the lowest energy states, illustrates a strong correspondence principle. It does not have the defects mentioned above and is performed without resorting to any probability axiom. The example suggests that the complex classical trajectories form the limiting case of the modified de Broglie-Bohm quantum trajectories.
http://arxiv.org/abs/1104.3197
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
David Gomez-Ullate, Paolo Santini, Matteo Sommacal, Francesco Calogero
We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.
http://arxiv.org/abs/1104.2205
Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS); Exactly Solvable and Integrable Systems (nlin.SI)