When physics helps mathematics: calculation of the sophisticated multiple integral

TL;DR

This paper leverages a connection between quantum mechanics and classical mechanics to compute a complex multi-dimensional integral, introducing new methods and an integral representation for ere9zeta(2).

Contribution

It establishes a novel link between quantum and classical problems to evaluate a sophisticated multiple integral and introduces a new integral representation for ere9zeta(2).

Findings

Calculated a complex multiple integral using classical-quantum correspondence.

Developed a direct calculation method effective in low-dimensional cases.

Derived a new integral representation for ere9zeta(2).

Abstract

There exists a remarkable connection between the quantum mechanical Landau-Zener problem and purely classical-mechanical problem of a ball rolling on a Cornu spiral. This correspondence allows us to calculate a complicated multiple integral, a kind of multi-dimensional generalization of Fresnel integrals. A direct method of calculation is also considered but found to be successful only in some low-dimensional cases. As a byproduct of this direct method, an interesting new integral representation for $\zeta(2)$ is obtained.