Set-valued mappings specified by regularization of the Schrödinger equation with degeneration

V. Zh. Sakbaev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 4, pp. 683-699

Résumé

The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.

Bibliographie

[1] Fichera G., “On a unified theory of boundary value problems for elliptic-parabolic equations of second order”, Boundary Problems in Differential Equations, Univ. Wisconsin Press, Madison, 1960, 97–120 | MR

[2] Oleinik O. A., “O lineinykh uravneniyakh vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Matem. sb., 69(111):1 (1966), 111–140 | MR

[3] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogransloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1957), 3–122 | MR | Zbl

[4] Freidlin M. I., “O stokhasticheskikh uravneniyakh Ito i vyrozhdayuschikhsya ellipticheskikh uravneniyakh”, Izv. AN SSSR. Ser. Matem., 26 (1962), 653–676 | MR

[5] S. N. Bakhvalov (red.), Trudy C. H. Kruzhkova, Sbornik statei, Fizmatlit, M., 2000, 14–38; 39–45; 99–153; 287–316

[6] Zhikov V. V., “K probleme predelnogo perekhoda v divergentnykh neravnomerno ellipticheskikh uravneniyakh”, Funkts. analiz i ego prilozh., 35:1 (2001), 23–39 | MR | Zbl

[7] Zhikov V. V., “Zamechaniya o edinstvennosti resheniya zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka s mladshimi chlenami”, Funkts. analiz i ego prilozh., 38:3 (2004), 15–28 | MR | Zbl

[8] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[9] Sakbaev V. Zh., “O postanovke zadachi Koshi dlya uravneniya Shredingera, vyrozhdayuschegosya na poluprostranstve”, Zh. vychisl. matem. i matem. fiz., 42:11 (2002), 1700–1711 | MR | Zbl

[10] Sakbaev V. Zh., “O funktsionalakh na resheniyakh zadachi Koshi dlya uravneniya Shredingera s vyrozhdeniem na polupryamoi”, Zh. vychisl. matem. i matem. fiz., 44:9 (2004), 1654–1673 | MR | Zbl

[11] Gerard P., “Microlocal defect measures”, Commun's. Part. Different. Equat., 16:11 (1991), 1761–1794 | DOI | MR | Zbl

[12] Kuk P., Beskonechnye matritsy i prostranstva posledovatelnostei, Fizmatlit, M., 1960 | MR

[13] Kholevo A. C., Veroyatnostnye i statisticheskie aspekty kvantovoi mekhaniki, In-t kompyuternykh issl., M., Izhevsk, 2003

[14] Orlov Yu. N., Osnovy kvantovaniya vyrozhdennykh dinamicheskikh sistem, MFTI, M., 2004

[15] Kozlov V. V., “Dinamika sistem s neintegriruemymi svyazyami. Ch. III”, Vestn. MGU. Ser. 1. Matem., mekhan., 1983, no. 3, 102–111 | Zbl

[16] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR | Zbl

[17] Dell'Antonio G. F., “On the limits of sequences of normal states”, Commun's. Pure and Appl. Math., 20 (1967), 413–429 | MR

[18] Sakbaev V. Zh., “O dinamike veroyatnostnykh mer, porozhdennoi zadachei Koshi dlya uravneniya Shredingera s vyrozhdeniem na polupryamoi”, Nekotorye probl. fundamentalnoi i prikl. matem., MFTI, M., 2004, 119–143

[19] Kuratovskii K., Topologiya, v. 1, Mir, M., 1966 | MR

[20] Rid M., Saimon B., Sovremennye metody matematicheskoi fiziki, v. 1, Mir, M., 1977 | MR

[21] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[22] Tolstov G. P., Mera i integral, Nauka, M., 1976 | MR | Zbl