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Month March 2013

Non-Hermitian \(\cal PT\)-symmetric quantum mechanics of relativistic particles with the restriction of mass


The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1+ \gamma_5 m_2\) are investigated. It is shown that for a fermion theory with a \(\gamma_5\)-mass term, the limiting of the mass specter by the value \( m_{max}= {m_1}^2/2m_2\) takes place. In this case the different regions of the unbroken \(\cal PT\) symmetry may be expressed by means of the restriction of the physical mass \(m\leq m_{max}\). It should be noted that in the approach which was developed by C.Bender et al. for the \(\cal PT\)-symmetric version of the massive Thirring model with \(\gamma_5\)-mass term, the region of the unbroken \(\cal PT\)-symmetry was found in the form \(m_1\geq m_2\) \cite{ft12}. However on the basis of the mass limitation \(m\leq m_{max}\) we obtain that the domain \(m_1\geq m_2\) consists of two different parametric sectors: i) \(0\leq m_2 \leq m_1/\sqrt{2}\) -this values of mass parameters \(m_1,m_2\) correspond to the traditional particles for which in the limit \(m_{max}\rightarrow \infty\) the modified models are converting to the ordinary Dirac theory with the physical mass \(m\); ii)\(m_1/\sqrt{2}\leq m_2 \leq m_1\) – this is the case of the unusual particles for which equations of motion does not have a limit, when \(m_{max}\rightarrow \infty\). The presence of this possibility lets hope for that in Nature indeed there are some “exotic fermion fields”. As a matter of fact the formulated criterions may be used as a major test in the process of the division of considered models into ordinary and exotic fermion theories. It is tempting to think that the quanta of the exotic fermion field have a relation to the structure of the “dark matter”.
Quantum Physics (quant-ph); High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Can unavoided level crossing disguise phase transition?

Miloslav Znojil

The answer is yes. Via an elementary, exactly solvable crypto-Hermitian example it is shown that inside the interval of admissible couplings the innocent-looking point of a smooth unavoided crossing of the eigenvalues of Hamiltonian $H$ may carry a dynamically nontrivial meaning of a phase-transition boundary or “quantum horizon”. Passing this point requires a change of the physical Hilbert-space metric $\Theta$, i.e., a thorough modification of the form and of the interpretation of the operators of all observables.

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Non-Hermitian star-shaped quantum graphs

Miloslav Znojil

A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowed with certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.
Quantum Physics (quant-ph)

Nonlinear stationary states in PT-symmetric lattices

Panayotis G. Kevrekidis, Dmitry E. Pelinovsky, Dmitry Y.Tyugin

In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation.
Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)

Spectral singularities in PT-symmetric Bose-Einstein condensates

W. D. Heiss, H. Cartarius, G. Wunner, J. Main

We consider the model of a PT-symmetric Bose-Einstein condensate in a delta-functions double-well potential. We demonstrate that analytic continuation of the primarily non-analytic term \(|\psi|^2 \psi\) – occurring in the underlying Gross-Pitaevskii equation – yields new branch points where three levels coalesce. We show numerically that the new branch points exhibit the behaviour of exceptional points of second and third order. A matrix model which confirms the numerical findings in analytic terms is given.
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

Nonuniqueness of the C operator in PT-symmetric quantum mechanics

Carl M. Bender, Mariagiovanna Gianfreda

The C operator in PT-symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, \(C^2=1\), \([C,PT]=0\), and \([C,H]=0\). These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate C. The usual approach has been to express the Hamiltonian as \(H=H_0+\epsilon H_1\), and to seek a solution for C in the form \(C=e^Q P\), where \(Q=Q(q,p)\) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form \(Q=\epsilon Q_1+\epsilon^3 Q_3+\epsilon^5 Q_5+\ldots\). [In previous work it has always been assumed that the coefficients of even powers of $\epsilon$ in this expansion would be absent because their presence would violate the condition that \(Q(p,q)\) is odd in p.] In an earlier paper it was argued that the C operator is not unique because the perturbation coefficient \(Q_1\) is nonunique. Here, the nonuniqueness of C is demonstrated at a more fundamental level: It is shown that the perturbation expansion for Q actually has the more general form \(Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots\) in which {\it all} powers and not just odd powers of \(\epsilon\) appear. For the case in which \(H_0\) is the harmonic-oscillator Hamiltonian, \(Q_0\) is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)