May 2012
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Month May 2012

Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials

Bikashkali Midya

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre and Jacobi-type \(X_1\) exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: \(e^{-\alpha p} x e^{\alpha p} = x+ i \alpha\), to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Path Integral Solution of PT-/non-PT-Symmetric and non-Hermitian Hulthen Potential

N. Kandirmaz, R. Sever

The wave functions and the energy spectrum of PT-/non-PT-Symmetric and non-Hermitian Hulthen potential are of an exponential type and are obtained via the path integral. The path integral is constructed using parametric time and point transformation.

Scattering from a discrete quasi-Hermitian delta function potential

Amine B Hammou

Scattering from a discrete quasi-Hermitian delta function potential is studied and the metric operator is found. A generalized continuity relation in the physical Hilbert space \({\mathcal H}_{{\rm phys}}\) is derived and the probability current density is defined. The reflection \({\mathcal R}\) and transmission \({\mathcal T}\) coefficients computed with this current are shown to obey the unitarity relation \({\mathcal R}+{\mathcal T}=1\).

PT-symmetry and quantum graphs

Miloslav Znojil

A new exactly solvable model of a quantum system is proposed, living on an equilateral q-pointed star graph (q is arbitrary). The model exhibits a weak and spontaneously broken form of \({\cal PT}-\)symmetry, offering a straightforward generalization of one of the standard solvable square wells with \(q=2\) and unbroken \({\cal PT}-\)symmetry. The kinematics is trivial, Kirchhoff in the central vertex. The dynamics is one-parametric (viz., \(\alpha-\)dependent), prescribed via complex Robin boundary conditions (i.e., the interactions are non-Hermitian and localized at the outer vertices of the star). The (complicated, trigonometric) secular equation is shown reducible to an elementary and compact form. This renders the model (partially) exactly solvable at any \(q \geq 2\) — an infinite subset of the real roots of the secular equation proves q-independent and known (i.e., inherited from the square-well \(q=2\) special case). The systems with \(q=4m-2\) are found anomalous, supporting infinitely many (or, at m=1, one) additional real m-dependent and \(\alpha-\)dependent roots.
Quantum Physics (quant-ph)

Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry

Ali Mostafazadeh

A PT-symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of PT-symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of PT-symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non-PT-symmetric configurations. This signifies the existence of non-PT-symmetric CPA-lasers.
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Optics (physics.optics)

Universal spectral behavior of \(x^2(ix)^ε\) potentials

Carl M. Bender, Daniel W. Hook

The PT-symmetric Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) (\(\epsilon\) real) exhibits a phase transition at \(\epsilon=0\). When \(\epsilon\geq0$\) the eigenvalues are all real, positive, discrete, and grow as \(\epsilon\) increases. However, when \(\epsilon<0\) there are only a finite number of real eigenvalues. As \(\epsilon\) approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at \(\epsilon=-1\). In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians \(H^{(2n)}=p^{2n}+x^2(ix)^\epsilon\) (\(\epsilon\) real, n=1, 2, 3, …). The complex classical behaviors of these Hamiltonians are also examined.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Complex Trajectories in a Classical Periodic Potential

Alexander G. Anderson, Carl M. Bender

This paper examines the complex trajectories of a classical particle in the potential \(V(x)=-\cos(x)\). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories \(x(t)\) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position \(x(0)\) after some real time \(T\). The second class consists of trajectories for which there exists a real time \(T\) such that \(x(t+T)=x(t) \pm2 \pi\). These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Light Transport in Random Media with PT-Symmetry

Samuel Kalish, Zin Lin, Tsampikos Kottos

The scattering properties of randomly layered optical media with \({\cal PT}\)-symmetric index of refraction are studied using the transfer-matrix method. We find that the transmitance decays exponentially as a function of the system size, with an enhanced rate \(\xi_{\gamma}(W)^{-1}=\xi_0(W)^{-1}+\xi_{\gamma} (0)^{-1}\), where \(\xi_0(W)\) is the localization length of the equivalent passive random medium and \(\xi_{\gamma}(0)\) is the attenuation/amplification length of the corresponding perfect system with a \({\cal PT}\)-symmetric refraction index profile. While transmitance processes are reciprocal to left and right incident waves, the reflectance is enhanced from one side and is inversely suppressed from the other, thus allowing such \({\cal PT}\)-symmetric random media to act as unidirectional coherent absorbers.
Optics (physics.optics)

PT-symmetric noncommutative spaces with minimal volume uncertainty relations

Sanjib Dey, Andreas Fring, Laure Gouba

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

Bypassing the bandwidth theorem with PT symmetry

Hamidreza Ramezani, J. Schindler, F. M. Ellis, Uwe Guenther, Tsampikos Kottos

The beat time \({\tau}_{fpt}\) associated with the energy transfer between two coupled oscillators is dictated by the bandwidth theorem which sets a lower bound \({\tau}_{fpt}\sim 1/{\delta}{\omega}\). We show, both experimentally and theoretically, that two coupled active LRC electrical oscillators with parity-time (PT) symmetry, bypass the lower bound imposed by the bandwidth theorem, reducing the beat time to zero while retaining a real valued spectrum and fixed eigenfrequency difference \(\delta\omega\). Our results foster new design strategies which lead to (stable) pseudo-unitary wave evolution, and may allow for ultrafast computation, telecommunication, and signal processing.
Classical Physics (physics.class-ph)