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Generalized solution of Schrödinger equation with singular potential and initial data | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 250، دوره 13، شماره 1، خرداد 2022، صفحه 3093-3101 اصل مقاله (397.69 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.23609.2565 | ||
| نویسندگان | ||
| Abdelmjid Benmerrous* ؛ Lalla Saadia Chadli؛ Abdelaziz Moujahid؛ Elomari M'hamed؛ Said Melliani | ||
| Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, PO Box 532,23000 Beni Mellal, Morocco | ||
| تاریخ دریافت: 16 خرداد 1400، تاریخ بازنگری: 03 شهریور 1400، تاریخ پذیرش: 13 شهریور 1400 | ||
| چکیده | ||
| This paper proved the existence and uniqueness of the solution of the Schrödinger equation with singular potential and initial data in the Colombeau algebra $\mathcal{G}_e$. | ||
| کلیدواژه‌ها | ||
| Schrödinger equation؛ Singular potential؛ initial data؛ Colombeau algebra؛ potentials | ||
| مراجع | ||
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[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. | ||
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