Speaker
Description
We consider linear equations with shifts of the arguments on the rectangular lattice with small step $h$ in $\mathbb{R}^n$ and construct a version of the canonical operator providing semiclassical asymptotics for such equations. In the case of functions of continuous argument, equations with shifts can be written as $h$-pseudodifferential equations with symbols $2\pi$-periodic in the momenta. This representation can also be given meaning for functions of a discrete argument, although differentiation operators are not defined for lattice functions. The phase space of such equations is the product $\mathbb{R}^n\times T^n$. Developing Maslov's ideas, we construct a canonical operator on Lagrangian submanifolds of this phase space with values in the space of lattice functions. Compared with the classical version of the canonical operator and the new formulas recently introduced by S.Yu.Dobrokhotov, A.I.Shafarevich, and the author, the construction involves a number of new features.
As an example, we give equations on a two-dimensional lattice that arise in quantum theory (the Feynman checkers model) and in the problem on the propagation of wave packets on a homogeneous tree.
The talk is based on the results of joint work with V.L.Chernyshev (HSE University, Moscow) and A.V.Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS, Moscow). The author's research was supported by the Russian Foundation for Basic Research under project no. 21-51-12006.
