Examples of potentials with convergent Schwinger - DeWitt expansion

TL;DR

This paper investigates the convergence of the Schwinger-DeWitt expansion for specific potentials, showing it converges only at discrete coupling constants, which aligns with the quantized nature of charge and has implications for quantum field theory.

Contribution

It demonstrates convergence of the Schwinger-DeWitt expansion for a special class of potentials and links this to discrete coupling constants, offering a new perspective on asymptotic expansions.

Findings

Expansion converges for certain potentials used in many-body problems.

Convergence occurs only at specific discrete values of the coupling constant.

Divergent cases exhibit essential singularities at the origin.

Abstract

Convergence of the Schwinger --- DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown, that this expansion, which is in general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and besides, convergence takes place only for definite discrete values of the coupling constant. For other values of the charge divergent expansion determines the kernels having essential singularity at origin (beyond usual $\delta$-function). If one consider only this class of potentials then one can avoid many problems, connected with asymptotic expansions, and one get the theory with discrete values of the coupling constant that is in correspondence with discreteness of the charge in the nature. This approach can be transmitted into the quantum field theory.