Mathematical center in Akademgorodok invites you to take part in the on
February 4 at 4 pm Novosibirsk time (UTC+7).
The seminar's heads: Andrey E. Mironov, Iskander A. Taimanov, Huijun Fan, Eugene Zhang.
During the seminar,
Sergey Yu. Dobrokhotov, Anna V. Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS) will present their talk
"Real-valued semiclassical approximation for the asymptotics with complex-valued phases of the Hermitian type orthogonal polynomials", based on joint work with Aleksandr I. Aptekarev, Dmitry N. Tulyakov (Keldysh Institute of Applied Mathematics RAS).
We consider orthogonal polynomials H_{n1,n2} (z, a) determined by the recurrence relations:
We obtain uniform asymptotics of diagonal polynomials H_{n,n} (z,a) in the form of an Airy function for n>>1, which is a far-reaching generalization of the Plancherel-Rotach asymptotic formulas for Hermitian polynomials. We discuss one of the possible approaches which we call "real semiclassics for asymptotics with complex-valued phases". Introducing an artificial small parameter h=0(1/n) and a continuous function 褎(x,z,a) such that H(z,a) = p(kh,z,a), we reduce the described problem to a pseudo-differential equation for 9, where x is a variable and (z,a) are parameters. Seeking its solution in the WKB-form, one obtains the Hamilton-Jacobi equations with complex Hamiltonians connected with a third-order algebraic curve. This circumstance is the main difficulty of solving the problem and, as a rule, leads to the transition from the real variable x to the complex one. In this problem, we propose a different approach. We divide the pseudo-differential equation in question into two with the following properties. The symbol of the first equation is real, the corresponding phase is real and is defined globally for all x. The operator defining the second equation has a complex-valued symbol, (complex Hamiltonian). However, this equation can be approximated by two equations, one of which has asymptotics with a purely imaginary phase, and the symbol of the second is pure real and has the form cos p+V_0 (x) + h V_1 (x) + O(h^2). This ultimately allows us to represent the desired asymptotic uniformly through the Airy function of the complex argument.
To take part, please, on February 4 after 3:45 pm Novosibirsk time (UTC+7) connect to the Zoom conference via this link: .
You can also connect manually in the Zoom app using the Conference ID: 616 7691 2581 (password: 392426).
Please pay attention to the following rules of attending the online seminar:
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