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Month November 2010

Gegenbauer-solvable quantum chain model

Miloslav Znojil

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, …$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer’s ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Phys. Rev. A 82 (2010) 052113

CPT-symmetric discrete square well

Miloslav Znojil, Miloš Tater

A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $\lambda\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

Perfect transmission scattering as a PT-symmetric spectral problem

H. Hernandez-Coronado, D. Krejcirik, P. Siegl

Transmissions |T|^2 as a function of energy k2 for the step-like potential <i>v</i> with a = Pi/4, epsilon_1 = 0.2, epsilon_3 = 0.5, beta_3 = -100, beta_2 = 0, beta_1 = -120 (continuous red line), and beta_1 = -200 (dashed blue line). See [5] for animated plots of |T|^2 as a function of potential.We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schroedinger operators with complex Robin boundary conditions.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

PT-symmetric quantum state discrimination

Carl M. Bender, Dorje C. Brody, Joao Caldeira, Bernard K. Meister

Suppose that a system is known to be in one of two quantum states, $|\psi_1 > $ or $|\psi_2 >$. If these states are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement to determine with certainty which state the system is in. However, because a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states, it is always possible to choose this inner product so that the two states $|\psi_1 > $ and $|\psi_2 > $ are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single measurement.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Universal routes to spontaneous PT-symmetry breaking in non-hermitian quantum systems

Henning Schomerus

(a) Sketch of a nonhermitian PT-symmetric system, where a region with absorption rate μ (and mean level spacing, left) is coupled symmetrically via a tunnel barrier (supporting N channels with transmission probability T) to an amplifying region with a matching amplification rate (right). Below this, the scattering description of the system. (b) Two routes to spontaneous PT-symmetry breaking, depending on whether the hermitian limit μ = 0 is T -symmetric (orthogonal class displaying level crossings, left) or not (unitary class displaying avoided crossings, right). Shown are real eigenvalues of a random Hamiltonian H [Eq. (4)] as function of T for fixed μ = 0 (left of dashed line), and then as a function of μ for fixed T = 1 (right of dashed line). Complex-valued levels (formed by level coalescence at μ > 0) are not shown. Here μ0 = √N/2, and we set N = 10.PT-symmetric systems can have a real spectrum even when their Hamiltonian is non-hermitian, but develop a complex spectrum when the degree of non-hermiticity increases. Here we utilize random-matrix theory to show that this spontaneous PT-symmetry breaking can occur via two distinct mechanisms, whose predominance is associated to different universality classes. Present optical experiments fall into the orthogonal class, where symmetry-induced level crossings render the characteristic absorption rate independent of the coupling strength between the symmetry-related parts of the system.
Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Optics (physics.optics)

Supersymmetry and PT-Symmetric Spectral Bifurcation

Kumar Abhinav, Prasanta K. Panigrahi

Dynamical systems exhibiting both PT and Supersymmetry are analyzed in a general scenario. It is found that, in an appropriate parameter domain, the ground state may or may not respect PT-symmetry. Interestingly, in the domain where PT-symmetry is not respected, two superpotentials give rise to one potential; whereas when the ground state respects PT, this correspondence is unique. In both scenarios, supersymmetry and shape-invariance are intact, through which one can obtain eigenfunctions and eigenstates exactly. Our procedure enables one to generate a host of complex potentials which are not PT-symmetric, and can be exactly solved.
Quantum Physics (quant-ph)