Geodesic
Canonical Operator on Punctured Lagrangian Manifolds and Commutation with Pseudodifferential Operators: Local Theory
Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 733-751.

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Maslov's canonical operator on punctured Lagrangian manifolds provides a solution to the Cauchy problem with initial data concentrated near a point or a submanifold of positive codimension for equations and wave-type systems for which the roots of the characteristic equation have singularities such as nonsmoothness and/or change of multiplicities at zero values of momenta. The theory of the canonical operator on punctured Lagrangian manifolds was constructed in the article [1] by S. Yu. Dobrokhotov, A. I. Shafarevich, and the author, in which, however, the formula for commutation of the canonical operator with pseudodifferential operators was not given. This formula is proved in the present article; moreover, the construction of the canonical operator on punctured Lagrangian manifolds is presented in an equivalent, more convenient form. We restrict ourselves to the local theory (the precanonical operator, or the operator in a separate chart of the Lagrangian manifold corresponding to some nondegenerate phase function), since the transition to the global construction does not contain anything new compared to the standard case.
Keywords: punctured Lagrangian manifold, Maslov's canonical operator, wave-type equation, localized initial data, change of multiplicity at zero momentum. 
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V. E. Nazaikinskii. Canonical Operator on Punctured Lagrangian Manifolds and Commutation with Pseudodifferential Operators: Local Theory. Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 733-751. http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a7/

[1] S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Canonical operator on punctured Lagrangian manifolds”, Russ. J. Math. Phys., 28:1 (2021), 22–36 | DOI | MR | Zbl

[2] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965

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

[4] L. Hörmander, “Fourier integral operators I”, Acta Math., 127 (1971), 79–183 | DOI | MR

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

[6] V. P. Maslov, M. V. Fedoryuk, “Logarifmicheskaya asimptotika bystro ubyvayuschikh reshenii giperbolicheskikh po Petrovskomu uravnenii”, Matem. zametki, 45:5 (1989), 50–62 | MR | Zbl

[7] S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, A. I. Shafarevich, “Asimptoticheskie bystroubyvayuschie resheniya lineinykh strogo giperbolicheskikh sistem s peremennymi koeffitsientami”, Matem. zametki, 49:4 (1991), 31–46 | MR | Zbl

[8] S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Propagation of a linear wave created by a spatially localized perturbation in a regular lattice and punctured Lagrangian manifolds”, Russ. J. Math. Phys., 24:1 (2017), 127–133 | DOI | MR | Zbl

[9] S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, “Uniform formulas for the asymptotic solution of a linear pseudodifferential equation describing water waves generated by a localized source”, Russ. J. Math. Phys., 27:2 (2020), 185–191 | DOI | MR | Zbl

[10] S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Effektivnye asimptotiki reshenii zadachi Koshi s lokalizovannymi nachalnymi dannymi dlya lineinykh sistem differentsialnykh i psevdodifferentsialnykh uravnenii”, UMN, 76:5 (461) (2021), 3–80 | DOI | MR | Zbl

[11] S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Novye integralnye predstavleniya kanonicheskogo operatora Maslova v osobykh kartakh”, Izv. RAN. Ser. matem., 81:2 (2017), 53–96 | DOI | MR | Zbl

[12] V. E. Nazaikinskii, B. Yu. Sternin, V. E. Shatalov, Metody nekommutativnogo analiza, Tekhnosfera, M., 2002