Mathematical Physics and Harmonic Analysis Seminar

Date: April 26, 2024

Time: 1:50PM - 2:50PM

Location: Zoom

Speaker: Gihyun Lee, Ghent University

Title: A calculus for magnetic pseudodifferential super operators

Abstract: The time evolution of a physical state is determined by the Liouville equation $\frac{d\rho}{dt} = -\frac{i}{\hbar}(H\rho - \rho H)$ in quantum mechanics. Here $\rho$ is the density operator describing a given physical state and $H$ is the Hamiltonian of a given system. Here we can observe that the Liouville operator $\rho\mapsto L_H \rho := -\frac{i}{\hbar}(H\rho - \rho H)$ assigns linear operators to linear operators - physicists call such an operator a super operator. On the other hand, various kinds of pseudodifferential calculi has been developed in mathematics and applied to the study of PDE, geometry and mathematical physics. The main idea behind these theories of pseudodifferential calculi is to construct systematic ways of assigning linear operators to symbol functions, which enables us to translate properties of functions to properties of linear operators. In this talk, I will introduce a novel pseudodifferential calculus of super operators in the magnetic setting and explain how the Liouville super operator $L_H$ can be incorporated into this new pseudodifferential theory. Furthermore, the $L_2$-boundedness of pseudodifferential super operators will be discussed. Based on the joint work with M. Lein.