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Month January 2012

PT-symmetrically deformed shock waves

Andrea Cavaglia, Andreas Fring

We investigate for a large class of nonlinear wave equations, which allow for shock wave formations, how these solutions behave when they are PT-symmetrically deformed. For real solutions we find that they are transformed into peaked solutions with a discontinuity in the first derivative instead. The systems we investigate include the PT-symmetrically deformed inviscid Burgers equation recently studied by Bender and Feinberg, for which we show that it does not develop any shocks, but peaks instead. In this case we exploit the rare fact that the PT-deformation can be provided by an explicit map found by Curtright and Fairlie together with the property that the undeformed equation can be solved by the method of characteristics. We generalise the map and observe this type of behaviour for all integer values of the deformation parameter epsilon. The peaks are formed as a result of mapping the multi-valued self-avoiding shock profile to a multi-valued self-crossing function by means of the PT-deformation. For some deformation parameters we also investigate the deformation of complex solutions and demonstrate that in this case the deformation mechanism leads to discontinuties.

http://arxiv.org/abs/1201.5809
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

PT-symmetric quantum systems with positive P

Miloslav Znojil, Hendrik B. Geyer

A new version of PT-symmetric quantum theory is proposed and illustrated by an N-site-lattice Legendre oscillator. The essence of the innovation lies in the replacement of parity P (serving as an indefinite metric in an auxiliary Krein space) by its non-involutory alternative P(positive)=Q>0 playing the role of a positive-definite nontrivial metric in an auxiliary, redundant, unphysical Hilbert space. It is shown that the QT-symmetry of this form remains appealing and technically useful.

http://arxiv.org/abs/1201.5058
Quantum Physics (quant-ph)

Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity

Francisco Correa, Mikhail S. Plyushchay

We study a reflectionless PT-symmetric quantum system described by the pair of complexified Scarf II potentials mutually displaced in the half of their pure imaginary period. Analyzing the rich set of intertwining discrete symmetries of the pair, we find an exotic supersymmetric structure based on three matrix differential operators that encode all the properties of the system, including its reflectionless (finite-gap) nature. The structure we revealed particularly sheds new light on the splitting of the discrete states into two families, related to the bound and resonance states in Hermitian Scarf II counterpart systems, on which two different series of irreducible representations of sl(2,C) are realized.

http://arxiv.org/abs/1201.2750
High Energy Physics – Theory (hep-th)

Quantum inner-product metrics via recurrent solution of Dieudonne equation

Miloslav Znojil

A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.

http://arxiv.org/abs/1201.2263
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Ordinary versus PT-symmetric φ^3 quantum field theory

Carl M. Bender, V. Branchina, Emanuele Messina

A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric ig\phi^3 quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian H=p^2+ix^3, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian g\phi^3 quantum field theory with those of the PT-symmetric ig\phi^3 quantum field theory. It is shown that while the conventional g\phi^3 theory in d=6 dimensions is asymptotically free, the ig\phi^3 theory is like a g\phi^4 theory in d=4 dimensions; it is energetically stable, perturbatively renormalizable, and trivial.

http://arxiv.org/abs/1201.1244
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Nonequilibrium perturbation theory in Liouville-Fock space for inelastic electron transport

Alan A. Dzhioev, D. S. Kosov

We use super-fermion representation of quantum kinetic equation to develop nonequilibrium perturbation theory for inelastic electron current through quantum dot. We derive Lindblad type kinetic equation for an embedded quantum dot (i.e. a quantum dot connected to Lindblad dissipators through a buffer zone). The kinetic equation is converted to non-Hermitian field theory in Liouville-Fock space. The general nonequilibrium many-body perturbation theory is developed and applied to the quantum dot with electron-vibron and electron-electron interactions. Our perturbation theory becomes equivalent to Keldysh nonequilibrium Green’s functions perturbative treatment provided that the buffer zone is large enough to alleviate the problems associated with approximations of the Lindblad kinetic equation.

http://arxiv.org/abs/1201.1230
Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Orthogonal Polynomials and S-curves

E.A.Rakhmanov

This paper is devoted to a study of S-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property (S-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an S-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to S-curves or may be interpreted this way. Existence problem S-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case when both the set of singularities of the external field and the set of fixed points of a class of curves are small (in main case — finite). We also discuss various applications and connections of the theorem.

http://arxiv.org/abs/1112.5713
Complex Variables (math.CV); Mathematical Physics (math-ph)

The spectrum of the cubic oscillator

Vincenzo Grecchi, André Martinez

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,\(H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3\),for \(\beta\) in the cut plane \(\C_c:=\C\backslash (-\infty, 0)\). Moreover, we prove that the spectrum consists of the perturbative eigenvalues \(\{E_n(\beta)\}_{n\geq 0}\) labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all \(\beta\in\C_c\), \(E_n(\beta)\) can be computed as the Stieltjes-Pad\’e sum of its perturbation series at \(\beta=0\). This also gives an alternative proof of the fact that the spectrum of \(H(\beta)\) is real when \(\beta\) is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.

http://arxiv.org/abs/1201.2797
Mathematical Physics (math-ph); Spectral Theory (math.SP)