International Journal of Nonlinear Analysis and Applications

Generalized solution of Schrödinger equation with singular potential and initial data

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, PO Box 532,23000 Beni Mellal, Morocco

Abstract

This paper proved the existence and uniqueness of the solution of the Schrödinger equation with singular potential and initial data in the Colombeau algebra $\mathcal{G}_e$.

Keywords

[1] J. Bourgain, Global solutions of nonlinear Schr¨odinger equations, AMS, Colloquium Publications, vol. 46, 1999.
[2] J.F. Colombeau, Elementary Introduction in New Generalized Functions, North Holland, Amsterdam, 1985.
[3] J.F. Colombeau, New Generalized Function and Multiplication of Distribution, North Holland, Amsterdam /New York / Oxford, 1984.
[4] R. Hermann and M. Oberguggenberger, Ordinary Differential Equations and Generalized Functions, M. Grosser, G. H¨ormann, M.Kunzinger, M. Oberguggenberger (Eds.), Proc. Workshop: Nonlinear Theory of Generalized Functions, E.Schrodinger Inst, Vienna, October-December 1997, in: Research Notes in Mathematical Series, Chapman and Hall/CRC, 1999.
[5] D. Rajterc Ciric and M. Stojanovic, Convolution-type derivatives and transforms of Colombeau generalized stochastic processes, Integral Transforms Spec. Funct. 22(45) (2011) 319–326.
[6] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, NewYork, 1975.
[7] M. Stojanovic, Extension of Colombeau algebra to derivatives of arbitrary order Dα, α ∈ R+ ∪ {0}. Application to ODEs and PDEs with entire and fractional derivatives, Nonlinear Anal. 71(11) (2009) 5458–5475.
[8] M. Stojanovic, Fondation of the fractional calculus in generalized function algebras, Anal. Appl. 10(4) (2012) 439–467.
[9] M. Stojanovic, Nonlinear Schr¨odinger equation with singular potential and initial data, Nonlinear Anal. 64 (2006) 1460–1474.
[10] S. Nakamura, Lectures on Schr¨odinger Operators, Lectures given at the University of Tokyo, October 1992, February 1993.
[11] M. Oberguggenberger, Generalized functions in nonlinear models a survey, Nonlinear Anal. 47 (2001) 5049–5040.
[12] C.L. Zhi and B. Fisher, Several products of distributions on Rm, Proc. R. Soc. Lond. A 426 (1989) 425–439.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 3093-3101
  • Receive Date: 06 June 2021
  • Revise Date: 25 August 2021
  • Accept Date: 04 September 2021