Geodesic
Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schr\"odinger equation with delta potential
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 704-715.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe the semiclassical asymptotic behavior of the solution of the Cauchy problem for the Schrödinger equation with a delta potential localized on a surface of codimension 1. The Schrödinger operator with a delta potential is defined using the theory of extensions and is given by the boundary conditions on this surface. The initial data are selected as a narrow peak, which is a Gaussian packet localized in a small neighborhood of the point. To construct the asymptotics, we use the Maslov complex germ method. We describe the reflection of the complex germ from the carrier of the delta potential.
Keywords: Schrödinger equation with a delta potential, semiclassical asymptotics of solution, Maslov complex germ method. 
@article{CMFD_2022_68_4_a9,
     author = {A. I. Shafarevich and O. A. Shchegortsova},
     title = {Maslov complex germ and semiclassical contracted states in the {Cauchy} problem for the {Schr\"odinger} equation with delta potential},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {704--715},
     publisher = {mathdoc},
     volume = {68},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a9/}
}
TY  - JOUR
AU  - A. I. Shafarevich
AU  - O. A. Shchegortsova
TI  - Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schr\"odinger equation with delta potential
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2022
SP  - 704
EP  - 715
VL  - 68
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a9/
LA  - ru
ID  - CMFD_2022_68_4_a9
ER  - 
%0 Journal Article
%A A. I. Shafarevich
%A O. A. Shchegortsova
%T Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schr\"odinger equation with delta potential
%J Contemporary Mathematics. Fundamental Directions
%D 2022
%P 704-715
%V 68
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a9/
%G ru
%F CMFD_2022_68_4_a9
A. I. Shafarevich; O. A. Shchegortsova. Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schr\"odinger equation with delta potential. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 704-715. http://geodesic.mathdoc.fr/item/CMFD_2022_68_4_a9/

[1] Berezin F. A., Faddeev L. D., “Zamechanie ob uravnenii Shredingera s singulyarnym potentsialom”, Dokl. AN SSSR, 137:5 (1961), 1011–1014

[2] Maslov V. P., Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nauka, M., 1977 | MR

[3] Maslov V. P., Asimptoticheskie metody i teoriya vozmuschenii, Nauka, M., 1988

[4] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[5] Ratyu T., Filatova T. A., Shafarevich A. I., “Nekompaktnye lagranzhevy mnogoobraziya, sootvetstvuyuschie spektralnym seriyam operatora Shredingera s delta-potentsialom na poverkhnosti vrascheniya”, Dokl. RAN, 446:6 (2012), 618–620 | MR

[6] Filatova T. A., Shafarevich A. I., “Kvaziklassicheskie spektralnye serii operatora Shredingera s delta-potentsialom na pryamoi i na sfere”, Teor. mat. fiz., 164:2 (2010), 279–298

[7] Shafarevich A. I., Schegortsova O. A., “Kvaziklassicheskaya asimptotika resheniya zadachi Koshi dlya uravneniya Shredingera s delta-potentsialom, lokalizovannym na poverkhnosti korazmernosti 1”, Tr. MIAN, 310, 2020, 322–331

[8] Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H., Solvable models in quantum mechanics, AMS Chelsea Publ., Providence, 2005 | MR

[9] Albeverio S., Kurasov P., Singular perturbations of differential operators, Cambridge Univ. Press, Cambridge, 2000 | MR

[10] Kronig R. de L., Penney W. G., “Quantum mechanics of electrons in crystal lattices”, Proc. R. Soc. London Ser. A, 130 (1931), 499–513 | DOI

[11] Ratiu T. S., Suleimanova A. A., Shafarevich A. I., “Spectral series of the Schrödinger operator with delta-potential on a three-dimensional spherically symmetric manifold”, Russ. J. Math. Phys., 20:3 (2013), 326–335 | DOI | MR