Pseudodifferential operators and a~canonical operator in general symplectic manifolds

M. V. Karasev ; V. P. Maslov

Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 277-305.

Résumé

A calculus of $h$-pseudodifferential operators with symbols on $\mathfrak X$ is defined modulo $O(h^2)$ on a closed symplectic manifold $(\mathfrak X,\omega)$ under the condition that $[\omega]/(2\pi h)-\varkappa/4 \in H^2(\mathfrak X,\mathbf Z)$. The class $\varkappa\in H^2(\mathfrak X,\mathbf Z)$ is described. On Lagrangian submanifolds $\Lambda\subset\mathfrak X$ a class in $H^1(\Lambda,\mathbf U(1))$ obstructing the definition of a canonical operator on $\Lambda$ is found. It is shown that an analogus calculus of pseudodifferential operators can be constructed with respect to homogeneity from an action of the group $\mathbf R_+$ on $\mathfrak X$. Bibliography: 22 titles.

Bibliographie

[1] Karasev M. V., Maslov V. P., “Globalnye asimptoticheskie operatory regulyatornogo predstavleniya”, Dokl. AN SSSR, 257:1 (1981), 33–38 | MR | Zbl

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

[3] Czyz J., “On geometric quantization and its connections with the Maslov theory”, Repts. Math. Phys., 15:1 (1979), 57–97 | DOI | MR | Zbl

[4] Hess H., “On a geometric quantization scheme generalizing those of Kostant–Sourian and Czyz”, Lect. Notes Phys., 139, 1981, 1–35 | MR

[5] Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Stemheimer D., “Deformation theory and quantization”, Ann. of Phys. (USA), 111:1 (1978), 61–151 | DOI | MR

[6] Neroslavskii O., Vlasov A., “Sur les deformations de l'algebra des fonctions”, C.R. Acad. Sci., Ser. A, 293:1 (1981), 71–73 | MR

[7] Anderson R. F. V., “The Weyl functional calculus”, J. Funct. Anal., 4:2 (1969), 240–267 | DOI | MR | Zbl

[8] Hörmander L., “The Weyl calculus of pseudo-differential operators”, Comm. Pure Appl. Math., 32:3 (1979), 359–443 | DOI | MR | Zbl

[9] Maslov V. P., Operatornye metody, Nauka, M., 1973 | MR

[10] Karasev M. V., Zadachnik po operatornym metodam, MIEM, M., 1979

[11] Karasev M. V., “O veilevskom i uporyadochennom ischislenii nekommutiruyuschikh operatorov”, Matem. zametki, 26:6 (1979), 885–907 | MR | Zbl

[12] Karasev M. V., “Asimptoticheskii spektr i front ostsillyatsii dlya operatorov s nelineinymi kommutatsionnymi sootnosheniyami”, Dokl. AN SSSR, 243:1 (1978), 15–18 | MR | Zbl

[13] Arnold V. I., “O kharakteristicheskom klasse, vkhodyaschem v usloviya kvantovaniya”, Funkts. analiz i ego prilozh., 1:1 (1967), 1–14 | MR

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

[15] Kucherenko V. V., “Kvaziklassicheskaya asimptotika funktsii tochechnogo istochnika dlya statsionarnogo uravneniya Shredingera”, Teor. i matem. fiz., 1:3 (1969), 384–405 | MR

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

[17] Maslov V. P., Nazaikinskii V. E., “Asimptotiki dlya uravnenii s osobennostyami v kharakteristikakh, I”, Izv. AN SSSR. Ser. matem., 45:5 (1981), 1049–1087 | MR | Zbl

[18] Karasev M. V., Nazaikinskii V. E., “O kvantovanii bystroostsilliruyuschikh simvolov”, Matem. sb., 106:2 (1978), 183–214 | MR

[19] Khermander L., “Integralnye operatory Fure, I”, Matematika, 16:1 (1972), 17–61 ; 2, 79–136 | Zbl

[20] Boutet de Monvel L., Guillemin V., “The spectral theory of Toeplitz operators”, Annals of Mathematics Studies, 99, Princeton University Press, Princeton, NJ, 1981, 161 | MR | Zbl

[21] Weinstein A., “Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds”, Comm. Pure Appl. Math., 30:2 (1977), 265–271 | DOI | MR | Zbl

[22] Karasev M. V., Maslov V. P., “Kvantovanie simplekticheskikh mnogoobrazii s konicheskimi tochkami”, Teor. i matem. fiz., 53:3 (1982), 374–387 | MR | Zbl