# Approximation formulas for evolution semigroups

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3786

dept.math [at] unitn.it

**Luogo**: PovoZero, via Sommarive, 14 - Povo - Sala Seminari "-1"**Ore**: 16:00

**Relatore**:

**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 R(t) = exp(i(S(t) - I)), which is a new example of a connection 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 yield the construction of the solution the Cauchy problem for a Schrödinger 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 Schrödinger equation with arbitrary high derivatives and variable coefficients, and also to multi-dimensional Schrödinger equation with unbounded locally square integrable potential.

References

[1]I.D. Remizov. Quasi-Feynman formulas as method of obtaining the evolution operator for the Schrödinger 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 Schrödinger Equation. Potential Analysis (2018) https://doi.org/10.1007/s11118- 018-9735-1

**Referente**: Sonia Mazzucchi