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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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A Regularized Asymptotic Solution of the Cauchy Problem for the Nonhomogeneous Schroedinger Equation in the Quasiclassical Approximation in the Presence of a Turning Point of the Limit Operator

Author(s):

Alexander Georgievich Eliseev

National Research University "Moscow Power Engineering Institute"
Associate Professor, Department of Higher Mathematics

yeliseevag@mpei.ru

Pavel Vladimirovich Kirichenko

National Research University "Moscow Power Engineering Institute"
Senior Lecturer, Department of Higher Mathematics

kirichenkopv@mpei.ru

Abstract:

The article is devoted to the development of the regularization method by S.A. Lomov on singularly perturbed problems in the presence of spectral singularities of the limit operator. In particular, a regularized asymptotic solution is constructed for the singularly perturbed inhomogeneous Cauchy problem that arises in the quasiclassical approximation in the Schroedinger equation in the coordinate representation. The potential energy profile chosen in the paper leads to a singularity in the spectrum of the limit operator in the form of a <> turning point. Based on the ideas of asymptotic integration of problems with unstable spectrum, S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the indicated type of singularity is described in detail, this algorithm is substantiated, and an asymptotic solution of any order with respect to a small parameter is constructed.

Keywords

References:

  1. Lomov S. A., Safonov V. F. [Regularizations and asymptotic solutions for singularly perturbed problems with point singularities of the spectrum of the limit operator]. Ukrains'kyi Matematychnyi Zhurnal, 1984; vol. 36, no. 2: 172-180. (In Russ. )
  2. Eliseev A. G., Lomov S. A. [Theory of singular perturbations in the case of spectral singularities of the limit operator]. Matematicheskii sbornik, 1986; vol. 131, no. 173: 544-557. (In Russ. )
  3. Bobodzhanov A. A., Safonov, V. F. [Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernels]. Ufa Mathematical Journal, 2018; vol. 10, no. 2: 3-12. (In Russ. )
  4. Eliseev A. G., Ratnikova T. A. [Singularly perturbed Cauchy problem in the presence of a rational “simple” turning point]. Differencial'nie uravnenia i processy upravlenia , 2019, no. 3 (In Russ. ) Available at: https://diffjournal.spbu.ru/RU/numbers/2019.3/article.1.3.html
  5. Eliseev A. G. [Regularized solution of a singularly perturbed Cauchy problem in the presence of an irrational “simple” turning point]. Differencial'nie uravnenia i processy upravlenia , 2020, no. 2 (In Russ. ) Available at: https://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.2.html
  6. Kirichenko P. V. [Singularly perturbed Cauchy problem for a parabolic equation in the presence of a “weak” turning point of the limit operator]. Matematicheskie zametki SVFU, 2020; no. 3: 3-15. (In Russ. )
  7. Eliseev A. G., Kirichenko P. V. [Regularized asymptotics of the solution of a singularly perturbed Cauchy problem in the presence of a “weak” turning point of the limit operator. ]. Differencial'nie uravnenia i processy upravlenia , 2020, no. 1 (In Russ. ) Available at: https://diffjournal.spbu.ru/RU/numbers/2020.1/article.1.4.html
  8. Eliseev A. G. [An example of solving a singularly perturbed Cauchy problem for a parabolic equation in the presence of a “strong” turning point]. Differencial'nie uravnenia i processy upravlenia , 2022, no. 3 (In Russ. ) Available at: https://diffjournal.spbu.ru/RU/numbers/2022.3/article.1.4.html
  9. Landau L. D., Lifshitz E. M. Kurs teoreticheskoy fiziki, T. 3, Kvantovaya mekhanika (nerelyativistskaya teoriya) [Course of Theoretical Physics, Vol. 3, Quantum Mechanics (non-relativistic theory)]. Moscow, Fizmatlit Publ., 2008. 800 p
  10. Maslov V. P. Teoriya vozmushcheniy i asimptoticheskiye metody [Perturbation theory and asymptotic methods]. Moscow, MGU Publ., 1965. 312 p
  11. Maslov V. P., Fedoryuk M. V. Kvaziklassicheskoye priblizheniye dlya uravneniy kvantovoy mekhaniki [Quasi-classical approximation for the equations of quantum mechanics]. Moscow, Nauka Publ., 1986. 296 p
  12. Kucherenko V. V. [Asymptotics of solutions of the system A(x, -ih d/dx) as h->0 in the case of characteristics of variable multiplicity]. Mathematics of USSR-Izvestiya, 1974, vol. 38, no. 3: 625-662. (In Russ. )
  13. Mishchenko A. S., Sternin B. U., Shatalov V. E. Lagranzhevy mnogoobraziya i metod kanonicheskogo operatora [Lagrange manifolds and the canonical operator method]. Moscow, Nauka Publ., 1978. 350 p
  14. Karasev M. V., Maslov V. P. [Pseudodifferential operators and canonical operator in symplectic manifolds]. Mathematics of USSR-Izvestiya, 1983, vol. 47, no. 5: 999-1029. (In Russ. )
  15. Lomov S. A. Vvedenie v obshyj teorij singuliajrnikh vozmyshenii [Introduction to the General Theory of Singular Perturbations]. Moscow, Nauka Publ., 1981. 400 p
  16. Arnold V. I. [On matrices depending on parameters]. - UMN, 1971, vol. 26, no. 2 (158): 101-114. (In Russ. )
  17. Liouville, J. Second Memoire sur le developpement des fonctions ou parties de fonctions en series dont les divers termes sont assujetis a satisfaire a une meme equation differentielle du second ordre, contenant un parametre variable. — Journal de Mathematiques Pures et Appliquees, 1837, p. 16-35. (In French)
  18. El'sgol'ts L. E. Differentsial'nyye uravneniya i variatsionnoye ischisleniye [Differential Equations and the Calculus of Variations]. Moscow, Nauka Publ., 1969. 424 p

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