Miscellaneous Results on Path Integration

I. Rogachevskii

Christian Grosche

Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.

Walter Greiner, J. Reinhardt

André LeClair, Giuseppe Mussardo, Hubert Saleur +1 more

We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O(n) and minimal models.

Victor K. Kuetche, Thomas Bouetou Bouetou, Timoléon Crépin Kofané +2 more

In this article, we investigate the structure and dynamics of miscellaneous mixtures of Bose-Einstein condensates confined within a time-independent anisotropic parabolic trap potential. In the zero-temperature mean-field approximation leading to coupled Gross-Pitaevskii equations for the macroscopic wave functions of the condensates, we show that these equations can be mapped onto the higher-dimensional time-gated Manakov system up to a first-order of accuracy. Paying particular attention to two-species mixtures and looking forward deriving a panel of miscellaneous excitations to the above equations, we analyze the singularity structure of the system by means of Weiss et al.'s [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983); 25, 13 (1984).] methodology and provide its general Lax representation. As a result, we unearth a typical spectrum of localized and periodic coherent patterns while depicting elastic and nonelastic interactions among such structures alongside the splitting and resonance phenomena occurring during their motion.

Antônio Sérgio Teixeira Pires