Integrals of Products of Bessel Functions: An Insight from the Physics of Bloch Electrons

TL;DR

This paper explores the mathematical properties of Bessel function integrals and connects their vanishing conditions to physical phenomena in Bloch electrons, such as density of states and umklapp scattering, providing a physical interpretation.

Contribution

It offers a novel physical interpretation of the conditions under which integrals of Bessel functions vanish, linking mathematical properties to electron behavior in lattice systems.

Findings

Integrals vanish when the electron energy is outside the allowed band.

Vanishing occurs when umklapp scattering is kinematically forbidden.

The physical interpretation relates to density of states and Fermi surface size.

Abstract

Integrals of products of Bessel functions exhibit an intriguing feature: under certain conditions on the parameters specifying the integrand, they vanish identically. We provide a physical interpretation of this feature in the context of both single-particle and many-body properties of electrons on a lattice (``Bloch electrons''), namely, in terms of their density of states and umklapp scattering rate. (In an umklapp event, the change in the momentum of two colliding electrons is equal to a reciprocal lattice vector, which gives rise to a finite resistivity due to electron-electron interaction.) In this context, the vanishing of an integral follows simply from the condition that either the density of states vanishes due to the electron energy lying outside the band in which free propagation of electron waves is allowed, or that an umklapp process is kinematically forbidden due to the Fermi surface being smaller than a critical value.