A construction of the fundamental solution of the Schrödinger equation with a perturbed quadratic Hamiltonian. (25th January 2016)

• Record Type: Journal Article
• Authors: Duboscq, Romain
• Is Part Of: Nonlinearity. Volume 29:Number 2(2016:Feb.)
• Page Start: 566
• Page End: 602
• Publication Date: 2016-01-25
• Subjects: nonlinear Schrödinger equation -- stochastic partial differential equation -- Hamiltonian system -- bicharacteristics method -- Strichartz estimates -- Avron–Herbst formula; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/2/566
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: The aim of this paper is to build a solution to the linear Schrödinger equation with a perturbed quadratic Hamiltonian. The solution is given in the sense of Sussmann (1978 Ann. Probab .6 19–41) and the construction is based on the bicharacteristics method. This is made possible under some assumptions on the Hamiltonian and the regularity of the perturbative noise. Moreover, dispersive estimates and an Avron–Herbst formula are also given during the analysis of the Cauchy problem for nonlinear Schrödinger equations.