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Интеллектуальная Система Тематического Исследования НАукометрических данных |
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Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\dotsm$ be a semi-classical Hamiltonian on $T^*R^n$, and let f be a semi-classical distribution (the ``source") microlocalized on a Lagrangian manifold $\Lambda$ which intersects transversally along $L$ the flow-out $\Lambda_+$ of the Hamilton vector field $X_{H_0}$ in a non-critical energy surface $\Sigma$. The paradigm of this situation is when $\Lambda$ is the conormal bundle to $x = 0$ and $H_0(x,p) = p_n$. This was considered in [1] from the point of view of asymptotics with respect to smoothness. As in [2] we are interested in integral representations for the solution $u$, modulo $O(h^N)$, verifying Sommerfeld radiation condition at infinity, of the inhomogeneous PDE $H_w(x,hD_x;h)u = f$. Using Maslov canonical operator, we present $u$ as the sum of terms microlocalized respectively on: (i) $\Lambda\setminus L$; (ii) the flow-out of $L$ by $X_{H_0}$ in $\Sigma$ for short times (near-field); (iii) the flow-out of $L$ through $X_{H_0}$ in $\Sigma$ for large times (far-field). We give various applications: (i) $H_w(x,hD_x;h)$ is a geodesic flow and $\Lambda$ a cylinder, as in the case for Bessel beams [3]; (ii) $H_w(x,hD_x;h)$ is Laplace operator and $\Lambda$ is the conormal to an hypersurface, as for the diffusion by an antenna; (iii) $H$ is the Dirichlet-to-Neumann operator for the linear waves equations in a domain with a non-uniform bottom, where $\Lambda$ is the conormal bundle to a point [4]. We shall also discuss non-transverse (glancing) intersection, extending to the semi-classical case some results of [5]. References [1] R.B. Melrose, G.A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32(4), 483–519 (1979). [2] A. Anikin, S. Dobrokhotov, V. Nazaikinski, M. Rouleux, The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides, Doklady Math., 96(1), 406–410 (2017). [3] S. Dobrokhotov, V. Nazaikinskii, A. Shafarevich, New integral representations of Maslov canonical operator in singular charts, Izv. Math., 81(2), 286–328 (2017). [4] A. Anikin, S. Dobrokhotov, V. Nazaikinski, M. Rouleux, Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom, Proceedings of “Days of Diffraction 2017”, Saint-Petersburg, 18–23. [5] P. Laubin, B. Willems, Distributions associated to a 2-microlocal pair of Lagrangian manifolds, Comm. Part. Diff. Eq., 19(9, 10), 1581–1610 (1994).
№ | Имя | Описание | Имя файла | Размер | Добавлен |
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1. | Программа конференции | 18-DD.pdf | 669,6 КБ | 22 января 2020 [anikin83@inbox.ru] |