Levy flights in a box

TL;DR

This paper investigates quantum Levy processes within a confined space, developing a path integral approach to solve the eigenvalue problem and analyze the process's behavior under topological constraints.

Contribution

It introduces a path integral formalism for quantum Levy processes in a box, providing analytical solutions and extending to oscillating boundary conditions.

Findings

Eigenvalue problem for quantum Levy in a box is solved.

Analytical expression for the evolution operator is derived.

Path integral approach correctly reproduces local quantum mechanics limits.

Abstract

It is shown that a quantum L\'evy process in a box leads to a problem involving topological constraints in space, and its treatment in the framework of the path integral formalism with the L\'evy measure is suggested. The eigenvalue problem for the infinite potential well is properly defined and solved. An analytical expression for the evolution operator is obtained in the path integral presentation, and the path integral takes the correct limit of the local quantum mechanics with topological constraints. An example of the L\'evy process in oscillating walls is also considered in the adiabatic approximation.