Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics

The continuity equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.Comment: 16 pages, amssy

PT-symmetric square well and the associated SUSY hierarchies

The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength $Z$ lies below the critical value $Z_0^{\rm (crit)}$ where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of $\sec^2$-like potentials in the $Z \to 0$ limit. For $Z$ above $Z_0^{\rm (crit)}$, there is a rich diversity of SUSY hierarchies, including some with PT-symmetry breaking and some with partial PT-symmetry restoration.Comment: LaTeX, 18 pages, no figure; broken PT-symmetry case added (Sec. 6

Bridging the Gap Between the Mode Coupling and the Random First Order Transition Theories of Structural Relaxation in Liquids

A unified treatment of structural relaxation in a deeply supercooled glassy liquid is developed which extends the existing mode coupling theory (MCT) by incorporating the effects of activated events by using the concepts from the random first order transition (RFOT) theory. We show how the decay of the dynamic structure factor is modified by localized activated events (called instantons) which lead to the spatial reorganization of molecules in the region where the instanton pops up. The instanton vertex added to the usual MCT depicts the probability and consequences of such an event which can be derived from the random first order transition theory. The vertex is proportional to $exp(-A/s_{c})$ where $s_{c}$ is the configurational entropy. Close to the glass transition temperature, $T_{g}$, since $s_{c}$ is diminishing, the activated process slows beyond the time window and this eventually leads to an arrest of the structural relaxation as expected for glasses. The combined treatment describes the dynamic structure factor in deeply supercooled liquid fairly well, with a hopping dominated decay following the MCT plateau.Comment: 11 pages, 5 figures, 1 tabl

Dynamical Heterogeneity and the interplay between activated and mode coupling dynamics in supercooled liquids

We present a theoretical analysis of the dynamic structure factor (DSF) of a liquid at and below the mode coupling critical temperature $T_c$, by developing a self-consistent theoretical treatment which includes the contributions both from continuous diffusion, described using general two coupling parameter ($F_{12}$) mode coupling theory (MCT), and from the activated hopping, described using the random first order transition (RFOT) theory, incorporating the effect of dynamical heterogeneity. The theory is valid over the whole temperature plane and shows correct limiting MCT like behavior above $T_{c}$ and goes over to the RFOT theory near the glass transition temperature, $T_{g}$. Between $T_{c}$ and $T_{g}$, the theory predicts that neither the continuous diffusion, described by pure mode coupling theory, nor the hopping motion alone suffices but both contribute to the dynamics while interacting with each other. We show that the interplay between the two contributions conspires to modify the relaxation behavior of the DSF from what would be predicted by a theory with a complete static Gaussian barrier distribution in a manner that may be described as a facilitation effect. Close to $T_c$, coupling between the short time part of MCT dynamics and hopping reduces the stretching given by the F$_{12}$-MCT theory significantly and accelerates structural relaxation. As the temperature is progressively lowered below $T_c$, the equations yield a crossover from MCT dominated regime to the hopping dominated regime. In the combined theory the dynamical heterogeneity is modified because the low barrier components interact with the MCT dynamics to enhance the relaxation rate below $T_c$ and reduces the stretching that would otherwise arise from an input static barrier height distribution.Comment: 7 pages, 4 figure

New approach to (quasi)-exactly solvable Schrodinger equations with a position-dependent effective mass

By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schr\"odinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.Comment: 11 pages, no figur

Universal power law in the orientational relaxation in thermotropic liquid crystals

We observe a surprisingly general power law decay at short to intermediate times in orientational relaxation in a variety of model systems (both calamitic and discotic, and also discrete) for thermotropic liquid crystals. As all these systems transit across the isotropic-nematic phase boundary, two power law relaxation regimes, separated by a plateau, emerge giving rise to a step-like feature (well-known in glassy liquids) in the single-particle second-rank orientational time correlation function. In contrast to its probable dynamical origin in supercooled liquids, we show that the power law here can originate from the thermodynamic fluctuations of the orientational order parameter, driven by the rapid growth in the second-rank orientational correlation length.Comment: Submitted to Physical Review Letter

Comparison of the effects of three different (-)-hydroxycitric acid preparations on food intake in rats: response

A response to Louter-van de Haar J, Wielinga PY, Scheurink AJ, Nieuwenhuizen AG: Comparison of the effects of three different (-)-hydroxycitric acid preparations on food intake in rats. Nutr Metabol 2005, 2:2

Generating Complex Potentials with Real Eigenvalues in Supersymmetric Quantum Mechanics

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT non-symmetric and its known eigenvalue corresponds to an unbound state.Comment: 20 pages, Latex 2e + amssym + graphics, 2 figures, accepted in Int. J. Mod. Phys.