Bargmann transform and its applications to partial differential equations

Abstract

This thesis is devoted to the fundamental properties and applications of the Bargmann transform and the Fock-Segal-Bargmann space. The fundamental properties include unitarity and invertibility of the transformation in L2 spaces and embeddings of the Fock-Segal-Bargmann spaces in Lp for any p>0. Applications include the linear partial differential equations such as the time-dependent Schrödinger equation in harmonic potential, the diffusion equation in self-similar variables, and the linearized Korteweg-de Vries equation, and one nonlinear partial differential equation given by the Gross-Pitaevskii model for the rotating Bose-Einstein condensate. The main question considered in this work in the context of linear partial differential equation is whether the envelope of the Gaussian function remains bounded in the time evolution. We show that the answer to this question is positive for the diffusion equation, negative for the Schrödinger equation, and unknown for the Korteweg-de Vries equation. We also address the local and global well-posedness of the nonlocal evolution equation derived for the Bose-Einstein condensates at the lowest Landau level.