XXXVIII Workshop on Geometric Methods in Physics | 30.06-6.07.2019 |

VIII School on Geometry and Physics | 24-28.06.2019 |

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## Vladimir Nazaikinskii## Fock Quantization of Canonical Transformations and Semiclassical Asymptotics for Degenerate ProblemsThe linear theory of run-up of long waves on a shallow beach involves differential operators degenerating in a special way on the boundary of the domain where the problem is considered (e.g., the velocity in the wave equation vanishes on the boundary as the square root of the distance from the boundary). The construction of semiclassical asymptotics for such problems is given by a version of Maslov's canonical operator based on a peculiar phase space geometry and using the Hankel transform to express rapidly oscillating solutions near the boundary. Note that the Hankel transform arises here as the Fock quantization of a classical canonical transformation regularizing the Hamiltonian system associated with the problem. We use this example to discuss the approach to degenerate problems in which new classes of operators arise by quantization of degenerate classical objects. |

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University of Bialystok |

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