We construct a previously unknown \(E_2\)-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.

http://arxiv.org/abs/1412.2800
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We propose the notion of \(E_2\)-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.

http://arxiv.org/abs/1411.4300
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.

http://arxiv.org/abs/1409.3828
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field \(\phi\). In Euclidean space the Lagrangian of such a theory, \(L=\frac{1}{2}(\nabla\phi)^2−ig\phi \exp(ia\phi)\), is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the \(m\)th energy level in the \(n\)th sector is given by \(E_{m,n}∼(m+1/2)^2a^2/(16n^2)\).

http://arxiv.org/abs/1408.2432
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We propose a noncommutative version of the Euclidean Lie algebra \(E_2\). Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.

http://arxiv.org/abs/1407.8097
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schroedinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy eigenspectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.

http://arxiv.org/abs/1401.4426
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behaviour is governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator and the Poeschl-Teller potential.

http://arxiv.org/abs/1305.4619
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space \(\Lc^2(\Bbb R^2)\), but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.

http://arxiv.org/abs/1310.4775

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

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \(\epsilon\) the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \(\epsilon\).

http://arxiv.org/abs/1305.7107
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg’s uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti-PT-symmetric modification to overcome this shortcoming.

http://arxiv.org/abs/1302.4571

Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

We investigate properties of generalized time-dependent q-deformed coherent states for a noncommutative harmonic oscillator. The states are shown to satisfy a generalized version of Heisenberg’s uncertainty relations. For the initial value in time the states are demonstrated to be squeezed, i.e. the inequalities are saturated, whereas when time evolves the uncertainty product oscillates away from this value albeit still respecting the relations. For the canonical variables on a noncommutative space we verify explicitly that Ehrenfest’s theorem hold at all times. We conjecture that the model exhibits revival times to infinite order. Explicit sample computations for the fractional revival times and superrevival times are presented.

http://arxiv.org/abs/1211.4791
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.

http://arxiv.org/abs/1207.3303
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets are clearly identified.

http://arxiv.org/abs/1207.3297
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.

http://arxiv.org/abs/1205.2291
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and non-linear integrable field equations of Korteweg-de-Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models we provide three alternative deformations: A complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real valued field equations of non linear integrable systems and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of KdV-type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.

http://arxiv.org/abs/1204.2291
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

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)

We provide a general construction procedure for antilinearly invariant complex root spaces. The proposed method is generic and may be applied to any Weyl group allowing to take any element of the group as a starting point for the construction. Worked out examples for several specific Weyl groups are presented, focusing especially on those cases for which no solutions were found previously. When applied in the defining relations of models based on root systems this usually leads to non-Hermitian models, which are nonetheless physically viable in a self-consistent sense as they are antilinearly invariant by construction. We discuss new types of Calogero models based on these complex roots. In addition we propose an alternative construction leading to q-deformed roots. We employ the latter type of roots to formulate a new version of affine Toda field theories based on non-simply laced roots systems. These models exhibit on the classical level a strong-weak duality in the coupling constant equivalent to a Lie algebraic duality, which is known for the quantum version of the undeformed case.

http://arxiv.org/abs/1108.1719
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We investigate complex versions of the Korteweg-deVries equations and an Ito type nonlinear system with two coupled nonlinear fields. We systematically construct rational, trigonometric/hyperbolic, elliptic and soliton solutions for these models and focus in particular on physically feasible systems, that is those with real energies. The reality of the energy is usually attributed to different realisations of an antilinear symmetry, as for instance PT-symmetry. It is shown that the symmetry can be spontaneously broken in two alternative ways either by specific choices of the domain or by manipulating the parameters in the solutions of the model, thus leading to complex energies. Surprisingly the reality of the energies can be regained in some cases by a further breaking of the symmetry on the level of the Hamiltonian. In many examples some of the fixed points in the complex solution for the field undergo a Hopf bifurcation in the PT-symmetry breaking process. By employing several different variants of the symmetries we propose many classes of new invariant extensions of these models and study their properties. The reduction of some of these models yields complex quantum mechanical models previously studied.

http://arxiv.org/abs/1103.1832
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E(8)-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.

http://arxiv.org/abs/1010.2218
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)