Research seminar "One-parameter semigroups and Applications"

Approximations to solution to the heat equation on a parallelizable manifold is presented. This will be done by means of the Chernoff theorem and Chernoff functions.

The moment problem is concerned with building a measure with given integrals of monomials of the form xⁿ, called moments. The talk will be about one method of solving the moment problem and about the applications of this method to bulding Chernoff's approximations. In particular, about the obstacles that we overcame in constructing the Chernoff's function in the form of an integral over a measure having moments that depend on time and the coefficients of the generator of the approximated semigroup.

The moment problem is concerned with building a measure with given integrals of monomials of the form xⁿ, called moments. The talk will be about one method of solving the moment problem and about the applications of this method to bulding Chernoff's approximations. In particular, about the obstacles that we overcame in constructing the Chernoff's function in the form of an integral over a measure having moments that depend on time and the coefficients of the generator of the approximated semigroup.

The Chernoff theorem allows to build approximations to a strongly continuous operators semigroup if the Chernoff's function and semigroup's generetor are known. However, the convergence rate of these approximations depends on the initial condition. The report will be devoted to an apparatus adapted to study this dependence - proposed by I. D. Remizov, the concept of approximation subspace.

The talk is based on the following papers:
1. Pointwise convergence of solution to Schrödinger equation
on manifolds by Xing Wang and Chunjie Zhang.
2. On the Derivation of the Schroedinger Equation in a Riemannian Manifold by K. R. Parthasabathy.
3. The Schrödinger equation on a compact manifold: Strichartz estimates and applications by Nicolas Burq, Patrick Gérard and Nikolay Tzvetkov.

The talk will cover some aspects as the Laplace-Beltrami operator for a Riemannian manifold, establish the convergence of the Schrodinger solution on known surfaces such as sphere an torus, and global existence of solutions.

The talk will present a proof of the theorem that the rate of convergence in Chernoff's theorem can be arbitrarily slow (for example, a group of translations on the real line). It will also be shown that the rate of convergence can be infinitely large - it is enough to take the semigroup itself as a Chernov function.

The talk will be devoted to the formula recently obtained by the author, this formula provides fast-converging Chernoff approximations to solution of the heat equation with variable coefficient of thermal conductivity.