The positive temperature Gibbs state of a scalar boson field with a relativistic local self-interaction in two space-time dimensional Minkowski universe as constructed in ~1] is not relativistic invarianto We prove in this paper that the corresponding state in the De Sitter universe is actually 1 relativistic invariant if the temperature is given by T = 2~R where R is the constant radius of curvature of the De Sitter universe. Moreover the construction gives that the Schwinger functions or imaginary time Wightman functions are the moments of a generalized Markoff process on the sphere of radius R.
We consider the Schrödinger equation in R 3 with nonlinearity concentrated in a finite set of points. We formulate the problem in the space of finite energy V , which is strictly larger than the standard H 1 -space due to the specific singularity exhibited by the solutions. We prove local existence and, for a repulsive or weakly attractive nonlinearity, also global existence of the solutions. 2003 Éditions scientifiques et médicales Elsevier SAS Keywords: Nonlinear Schrödinger Equation (NLSE); Point interactions; Nonlinear Dirac Delta potentials; Existence and uniqueness in energy space RÉSUMÉ. -On considère l'équation de Schrödinger avec une nonlinéarité concentrée en un nombre fini de points. On formule le problème dans l'espace V d'énergie finie, qui contient strictement l'espace standard H 1 , à cause de la singularité spécifique des solutions. On prouve des résultats d'existence locale et même, pour une nonlinéarité répulsive ou faiblement attractive, l'existence globale des solutions. 2003 Éditions scientifiques et médicales Elsevier SAS
We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the "strength" of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t → ∞, starting from a bound state at time t = 0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states.
Abstract. After discussing in several contexts how mean field and fluctuation approximations arise and can be used, we give a simple method by which the analysis of the approximations can be carried out.
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