Asymptotic distribution of eigenvalues

S. Z. Levendorskii

Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 119-160.

Résumé

The method of a spectral almost-projection is employed to investigate the masymptotic behavior of the eigenvalue distribution functions of semibounded essentially selfadjoint systems of differential operators in a domain and of pseudodifferential operators on $R^n$. Use is made of the calculus of pseudodifferential operators with dual symbols constructed in the article. Bibliography: 27 titles.

Bibliographie

[1] Tulovskii V. N., Shubin M. A., “Ob asimptoticheskom raspredelenii sobstvennykh znachenii psevdodifferentsialnykh operatorov v $R^n$”, Matem. sb., 92:4 (1973), 571–578 | MR

[2] Feigin V. I., “Novye klassy psevdodifferentsialnykh operatorov v $R^n$ i nekotorye prilozheniya”, Trudy Mosk. matem. ob-va, 36, 1978, 155–194 | MR | Zbl

[3] Feigin V. I., “Asimptoticheskoe raspredelenie sobstvennykh znachenii dlya gipoellipticheskikh sistem v $R^n$”, Matem. sb., 99:4 (1976), 594–614 | MR | Zbl

[4] Roitburd V. L., “O kvaziklassicheskoi asimptotike spektra psevdodifferentsialnogo operatora”, UMN, 31:4 (1976), 275–276 | MR | Zbl

[5] Feigin V. I., “Asimptoticheskoe raspredelenie sobstvennykh chisel i formula tipa Bora–Zommerfelda”, Matem. sb., 110:1 (1979), 66–87 | MR | Zbl

[6] Levendorskii S. Z., “Algebry psevdodifferentsialnykh operatorov s razryvnymi simvolami”, Dokl. AN SSSR, 248:4 (1979), 777–779 | MR | Zbl

[7] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978 | MR

[8] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra differentsialnykh uravnenii”, Itogi nauki. Matematicheskii analiz, 14, 1977, 5–58 | MR | Zbl

[9] Boimatov K. X., Kostyuchenko A. G., “Raspredelenie sobstvennykh znachenii ellipticheskikh operatorov vo vsem prostranstve”, Trudy sem. im. I. G. Petrovskogo, 2, 1976, 113–142 | MR

[10] Boimatov K. X., “Raspredelenie sobstvennykh znachenii ellipticheskikh operatorov”, Dokl. AN SSSR, 221:2 (1975), 265–268 | MR | Zbl

[11] Boimatov K. X., “Spektralnye asimptotiki differentsialnykh uravnenii s operatornymi koeffitsientami v negladkoi oblasti”, Dokl. AN SSSR, 242:4 (1978), 749–752 | MR | Zbl

[12] Boimatov K. X., “Psevdodifferentsialnye operatory s operatornym simvolom”, Dokl. AN SSSR, 244:1 (1979), 20–23 | MR | Zbl

[13] Markus A. S., Matsaev V. I., “Ob asimptotike spektra operatorov, blizkikh k normalnym”, Funkts. analiz i ego pril., 13:3 (1979), 93–94 | MR | Zbl

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

[15] Mischenko A. S., Sternin B. Yu., Shatalov V. E., Lagranzhevy mnogoobraziya i metod kanonicheskogo operatora, Nauka, M., 1978 | MR

[16] Vishik M. I., Grushin V. V., “Kraevye zadachi dlya ellipticheskikh uravnenii, vyrozhdayuschikhsya na granitse oblasti”, Matem. sb., 80:4 (1969), 455–491 | Zbl

[17] Levendorskii S. Z., “Kraevye zadachi v poluprostranstve dlya kvaziellipticheskikh differentsialnykh operatorov, vyrozhdayuschikhsya na granitse”, Matem. sb., 111:4 (1980), 483–502 | MR | Zbl

[18] Glushko V. P., Batalin M. D., “Koertsitivnaya razreshimost obschikh granichnykh zadach dlya nekotorykh vyrozhdayuschikhsya ellipticheskikh uravnenii vysokogo poryadka”, Teoriya kubaturnykh formul i prilozheniya funktsionalnogo analiza k nekotorym zadacham matematicheskoi fiziki, Novosibirsk, 1975

[19] Feigin V. I., “O nëterovosti differentsialnykh operatorov v $R^n$”, Differentsialnye uravneniya, 11:12 (1975), 2231–2235 | MR | Zbl

[20] Feigin V. I., “O spektralnoi asimptotike dlya kraevykh zadach i asimptotike otritsatelnogo spektra”, Dokl. AN SSSR, 232:6 (1977), 1269–1272 | MR

[21] Bezyaev V. I., “Asimptotika sobstvennykh znachenii gipoellipticheskikh operatorov na zamknutom mnogoobrazii”, Dokl. AN SSSR, 244:5 (1979), 1054–1057 | MR | Zbl

[22] Beats R., “A general calculus of pseudodifferential operators”, Duke Math. J., 42:1 (1975), 1–42 | DOI | MR

[23] Hörmander L., “The Weyl calculus of pseudodifferential operators”, Comm. Pure Appl. Math., 32 (1979), 359–443 | DOI | MR | Zbl

[24] Robert D., “Proprietes spectrales d'operateurs pseudodifferentiels”, Comm. Part. Differ. Equat., 3:9 (1978), 755–826 | DOI | MR | Zbl

[25] Hörmander L., “On the asymptotic distribution of the eigenvalues of pseudodifferential operators in $R^n$”, Ark. Mat., 17:2 (1979), 297–313 | DOI | MR | Zbl

[26] Knapp A., Stein E. M., “Singular integrals and principal series”, Proc. Nat. Acad. Sci. USA, 63 (1969), 297–313 | DOI

[27] Calderon A. P., Vaillancourt R., “On the boundedness of pseudodifferential operators”, J. Math. Soc. Japan, 32 (1971), 374–378 | MR