Generalized solution to multidimensional cubic Schrödinger equation with delta potential

Abstract

© 2019, Springer-Verlag GmbH Austria, part of Springer Nature. This article addresses the Cauchy problem for the defocusing cubic Schrödinger equation in 2D and 3D and the equation with a delta well potential in 3D. Solutions belong to the Colombeau algebra of generalized functions GC1,H2 (see Nedeljkov et al. in Proc R Soc Edinb Sect A Math 135:863–886, 2005). The physically significant homogeneous problem in 2D and 3D has not yet been treated in this framework, whereas no classical results exist on the equation with delta potential. The paper contains the construction of unique generalized solutions for both of these problems. One could also find an assertion about compatibility with classical solutions for 2D and 3D equations without delta potential.