U. D. Jentschura, B. J. Wundt
We show that it is possible to construct a tachyonic version of the Dirac equation, which contains the fifth current and reads (i gamma^\mu partial_\mu – gamma^5 m) \psi = 0. Its spectrum fulfills the dispersion relation E^2 = p^2 – m^2, where E is the energy, p is the spatial momentum, and m is the mass of the particle. The tachyonic Dirac equation is shown to be CP invariant, and T invariant. The Feynman propagator is found. In contrast to the covariant formulation, the tachyonic Hamiltonian H_5 = alpha.p + beta gamma^5 m breaks Lorentz covariance (as does the Hamiltonian formalism in general, because the time variable is singled out and treated differently from space). The tachyonic Dirac Hamiltonian H_5 breaks parity but is found to be invariant under the combined action of parity and a noncovariant time reversal operation T’. In contrast to the Lorentz-covariant T operation, T’ involves the Hermitian adjoint of the Hamiltonian. Thus, in the formalism developed by Bender et al., primarily in the context of quantum mechanics, H_5 is PT’ symmetric. The PT’ invariance (in the quantum mechanical sense) is responsible for the fact that the energy eigenvalues of the tachyonic Dirac Hamiltonian are real rather than complex. The eigenstates of the Hamiltonian are shown to approximate the helicity eigenstates of a Dirac neutrino in the massless limit.
http://arxiv.org/abs/1110.4171
High Energy Physics – Phenomenology (hep-ph)
Philip Cherian, Kumar Abhinav, P. K. Panigrahi
A family of PT -symmetric complex potentials are obtained which is isospectral to free particle in an infinite complex box in one dimension (1-D). These are generalizations to the cosec2(x) potential, isospectral to particle in a real infinite box. In the complex plane, the infinite box is extended parallel to the real axis having a real width, which is found to be an integral multiple of a constant quantum factor, arising due to boundary conditions necessary for maintaining the PT -symmetry of the superpartner. As the spectra of the particle in a box is still real, it necessarily picks out the unbroken PT -sector of its superpartner, thereby invoking a close relation between PT -symmetry and SUSY for this case.
http://arxiv.org/abs/1110.3708
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
R. Driben, B. A. Malomed
We introduce a system based on dual-core nonlinear waveguides with the balanced gain and loss acting separately in the cores. The system features a “supersymmetry” when the gain and loss are equal to the inter-core coupling. This system admits a variety of exact solutions (we focus on solitons), which are subject to a specific subexponential instability. We demonstrate that the application of a “management”, in the form of periodic simultaneous switch of the sign of the gain, loss, and inter-coupling, effectively stabilizes solitons, without destroying the supersymmetry. The management turns the solitons into attractors, for which an attraction basin is identified. The initial amplitude asymmetry and phase mismatch between the components transforms the solitons into quasi-stable breathers.
http://arxiv.org/abs/1110.2409
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Jun-Qing Li, Yan-Gang Miao
By adding an imaginary potential proportional to \(ip_1p_2\) to the hamiltonian of an anisotropic planar oscillator, we construct a model which is described by a non-hermitian hamiltonian with PT pseudo-hermiticity. We introduce the mechanism of the spontaneous breaking of permutation symmetry of the hamiltonian for diagonalizing the hamiltonian. By applying the definition of annihilation and creation operators which are PT pseudo-hermitian adjoint to each other, we give the real spectra.
http://arxiv.org/abs/1110.2312
Quantum Physics (quant-ph)
Jun-Qing Li, Yan-Gang Miao
By adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a model which is described by a non-hermitian hamiltonian with PT pseudo-hermiticity. We introduce the mechanism of the spontaneous breaking of permutation symmetry of the hamiltonian for diagonalizing the hamiltonian. By applying the definition of annihilation and creation operators which are PT pseudo-hermitian adjoint to each other, we give the real spectra.
http://arxiv.org/abs/1110.2312
Quantum Physics (quant-ph)
Gilles Demange, Eva-Maria Graefe
Parameter dependent non-Hermitian quantum systems typically not only possess eigenvalue degeneracies, but also degeneracies of the corresponding eigenfunctions at exceptional points. While the effect of two coalescing eigenfunctions on cyclic parameter variation is well investigated, little attention has hitherto been paid to the effect of more than two coalescing eigenfunctions. Here a characterisation of behaviours of symmetric Hamiltonians with three coalescing eigenfunctions is presented, using perturbation theory for non-Hermitian operators. Two main types of parameter perturbations need to be distinguished, which lead to characteristic eigenvalue and eigenvector patterns under cyclic variation. A physical system is introduced for which both behaviours might be experimentally accessible.
http://arxiv.org/abs/1110.1489
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Sergey V. Suchkov, Boris A. Malomed, Sergey V. Dmitriev, Yuri S. Kivshar
Dynamics of a chain of interacting parity-time invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrodinger (DNLS) equation is demonstrated. Approximate solutions for solitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart of the discrete equations. These solitons are mobile, featuring nearly elastic collisions. Stationary solutions for narrow solitons, which are immobile due to the pinning by the effective Peierls-Nabarro potential, are constructed numerically, starting from the anti-continuum limit. The solitons with the amplitude exceeding a certain critical value suffer an instability leading to blowup, which is a specific feature of the nonlinear PT-symmetric chain, making it dynamically different from DNLS lattices. A qualitative explanation of this feature is proposed. The instability threshold drops with the increase of the gain-loss coefficient, but it does not depend on the lattice coupling constant, nor on the soliton’s velocity.
http://arxiv.org/abs/1110.1501
Optics (physics.optics)
Joe Watkins
We study the spectral zeta functions associated to the radial Schrodinger problem with potential \(V(x)=x^{2M}+\alpha x^{M-1} +(\lambda^2-1/4)/x^2\). Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular \({}_5F_4\) hypergeometric series as an example. Our work is then extended to a class of related \({\cal PT}\)-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion \(G_n\) which appear in the associated integrable quantum field theory.
http://arxiv.org/abs/1110.2004
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th)
Miloslav Znojil
An extension of the scope of quantum theory is proposed in a way inspired by the recent heuristic as well as phenomenological success of the use of non-Hermitian Hamiltonians which are merely required self-adjoint in a Krein space with an indefinite metric (chosen, usually, as the operator of parity). In nuce, the parity-like operators are admitted to represent the mere indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and, via a non-numerical illustrative example, found feasible.
http://arxiv.org/abs/1110.1218
Mathematical Physics (math-ph)
Oleg N. Kirillov
When gain and loss are in perfect balance, dynamical systems with indefinite damping can obey the exact PT-symmetry and therefore be marginally stable with a pure imaginary spectrum. At an exceptional point where the exact PT-symmetry is spontaneously broken, the stability is lost via a Krein collision of eigenvalues just as it happens at the Hamiltonian Hopf bifurcation. In the parameter space of a general dissipative system, marginally stable PT-symmetric ones occupy singularities on the boundary of the asymptotic stability domain. To observe how the singular surface governs dissipation-induced destabilization of the PT-symmetric system when gain and loss are not matched, an extension of recent experiments with PT-symmetric LRC circuits is proposed.
http://arxiv.org/abs/1110.0018
Mathematical Physics (math-ph); Other Condensed Matter (cond-mat.other); Spectral Theory (math.SP); Quantum Physics (quant-ph)
Rodislav Driben, Boris A. Malomed
Families of analytical solutions are found for symmetric and antisymmetric solitons in the dual-core system with the Kerr nonlinearity and PT-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the “supersymmetric” case, with equal coefficients of the gain, loss and inter-core coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching (“management”).
http://arxiv.org/abs/1109.5759
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)