Approximation formulas for evolution semigroups

Venue:  PovoZero, via Sommarive, 14 - Povo - Sala Seminari "-1"
At: 4:00 pm

Speaker: 

• Ivan  D. Remizov - HSE University, Russia

Abstract:

The talk is devoted to some recent results by Prof. Remizov [1, 2, 3, 4], and will include:

First, Remizov’s formula R(t) = exp(i(S(t)  I)), which is a new example of a connec-  tion between semigroups with generators H and iH. The classical notion of Chernoff- equivalent family of operators is replaced with the, less restrictive, notion of Chernoff- tangent family. This is the technique which allows to yild the construction of the solution the Cauchy problem for a Schr¨odinger equation from the construction of a less difficult Chernoff-tangent family for a less difficult heat equation [1].
Second, the idea of using shift operators instead of integral operators while constructing solutions for parabolic heat-type equations [2, 3].
Third, the synthesis of the above ideas that allow [4] to construct solutions for one- dimensional Schr¨odinger equation with arbitrary high derivatives and variable coefficients, and also to multi-dimensional Schr¨odinger equation with unbounded locally square inte- grable potential.

References
[1]I.D. Remizov. Quasi-Feynman formulas as method of obtaining the evolution operator for the Schr¨odinger equation.// J. Funct. Anal. 270 (2016), no. 12, 4540-4557.
[2]I.D. Remizov. Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example).// Appl. Math. Comput. 328 (2018), 243-246.
[3]I.D. Remizov. Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients// arXiv:1710.06296 (Manuscript submitted to Journal of Mathematical Physics)
[4]I.D. Remizov. Formulas that Represent Cauchy Problem Solution for Momentum and Po- sition Schr¨odinger Equation// Potential Analysis (2018) https://doi.org/10.1007/s11118- 018-9735-1

Contact person: Sonia Mazzucchi