Semigroups for quadratic evolution equations acting on Shubin-Sobolev and Gelfand-Shilov spaces

TL;DR

This paper studies the solution semigroup for quadratic evolution equations with non-negative definite Hamiltonians, demonstrating strong continuity across various functional spaces including Shubin--Sobolev, Schwartz, and Gelfand--Shilov spaces.

Contribution

It establishes the strong continuity of the solution semigroup on multiple advanced function spaces for a class of quadratic evolution equations.

Findings

Strong continuity on Shubin--Sobolev spaces

Extension to Schwartz and Gelfand--Shilov spaces

Applicability to ultradistribution spaces

Abstract

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin--Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand--Shilov spaces and their dual ultradistribution spaces.