On the asymptotic behavior of the spectral characteristics of exterior problems for the Schr\"odinger operator

V. S. Buslaev

Izvestiya. Mathematics , Tome 9 (1975) no. 1, pp. 139-223.

Résumé

The Green's function $G(x,y;\lambda)$, $x,y\in\Omega$, $\lambda>0$, of the Schrödinger equation $-\Delta_xG+v(x)G-\lambda G=\delta(x-y)$ satisfying a radiation condition at infinity is considered in the exterior $\Omega$ of a convex smooth closed hypersurface $\Gamma$ in $R^m$. The potential is assumed to be a smooth function with compact support. Asymptotic formulas for $\lambda\to\infty$that are uniform in $x$ and $y$ are obtained. Bibliography: 17 items.

Bibliographie

[1] Buslaev V. S., “Formuly sledov dlya operatora Shredingera v trekhmernom prostranstve”, Dokl. AN SSSR, 143:5 (1962), 1067–1070 | MR | Zbl

[2] Buslaev V. S., “Formuly sledov i nekotorye asimptoticheskie otsenki yadra rezolventy dlya operatora Shredingera v trekhmernom prostranstve”, Problemy matem. fiziki, no. 1, LGU, 1966, 82–101 | MR

[3] Arsenev A. A., “Asimptotika spektralnoi funktsii uravneniya Shredingera”, Zh. vychisl. matem. i matem. fiz., 7:6 (1967), 507–518 | MR

[4] Buslaev V. S., “Rasseyannye ploskie volny, spektralnye asimptotiki i formuly sleda dlya vneshnikh zadach”, Dokl. AN SSSR, 12:2 (1971), 999–1002 | MR | Zbl

[5] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, MGU, 1965

[6] Kucherenko V. V., “Kvaziklassicheskaya asimptotika funktsii tochechnogo istochnika dlya statsionarnogo uravneniya Shredingera”, Teoretich. i Matem. fizika, 1:3 (1969), 384–406 | MR

[7] Buslaev V. S., “Korotkovolnovaya asimptotika v zadache difraktsii na gladkikh vypuklykh konturakh”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 73, 1964, 14–117 | MR

[8] Babich V. M., “O korotkovolnovoi asimptotike funktsii Grina dlya uravneniya Gelmgoltsa”, Matem. sb., 65:4 (1964), 571–573

[9] Filippov V. B., “O strogom opravdanii korotkovolnovoi asimptotiki dlya zadachi difraktsii v zone teni”, Matem. voprosy teorii raspr. voln, Zap. nauchn. semin. LOMI, 5, 1975, 142–205 | MR

[10] Morawetz C., Ludwig D., “An enequality for the reduced wave operator and the justification of geometrical optics”, Comm. Pure and Appl. Math., 21:2 (1968), 187–203 | DOI | MR | Zbl

[11] Babich V. M., Buldyrev V. S., Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[12] Buslaev V. S., “Teoriya potentsiala i geometricheskaya optika”, Zap. nauchn. semin. LOMI, 22, 1971, 175–180 | MR | Zbl

[13] Buslaev V. S., “Korotkovolnovaya asimptotika v okrestnosti raspolozhennogo na granitse istochnika”, VI Vsesoyuzn. simp. po difr. i raspr. voln, kn. 1, Moskva, Erevan, 1973, 37–40

[14] Babich V. M., “O strogom opravdanii korotkovolnovogo priblizheniya v trekhmernom sluchae”, Matemat. voprosy teorii raspr. voln. V, Zap. nauchn. semin. LOMI, 34, 1973, 23–51 | MR | Zbl

[15] Babich V. M., Grigoreva N. S., “Trekhmernyi analog metoda Vatsona”, VI Vsesoyuzn. simp. po difr. i raspr. voln, kn. II, Moskva, Erevan, 1973, 235–238

[16] Babich V. M., “O korotkovolnovoi asimptotike resheniya zadachi o tochechnom istochnike v neodnorodnoi srede”, Zh. vychisl. matem. i matem. fiz., 5:5 (1965), 949–951 | Zbl

[17] Fok V. A., Problemy difraktsii i rasprostraneniya elektromagnitnykh voln, Sovetskoe radio, 1970