The Liouville equation for singular ergodic magnetic Schr\"odinger operators

TL;DR

This paper investigates the time evolution of quantum states under ergodic magnetic Schrödinger operators with singular potentials, constructing a unitary propagator and solving the associated Liouville equation in a specialized Hilbert space.

Contribution

It introduces a method to handle the dynamics of quantum systems with singular magnetic and electric potentials, including the construction of a unitary propagator and solving the Liouville equation.

Findings

Successfully constructed a unitary propagator for the system.

Solved the Liouville equation in an appropriate Hilbert space.

Provided a framework for analyzing quantum dynamics with singular potentials.

Abstract

We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schr\"odinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schr\"odinger equation, and solve a Liouville equation in an appropriate Hilbert space.