Analytical Properties of Solutions of the Schrodinger Equation and Quantization of Charge

TL;DR

This paper investigates the analytical properties of solutions to the Schrödinger equation using the Schwinger--DeWitt expansion, revealing conditions for convergence and the quantization of charge in certain potentials.

Contribution

It demonstrates that the Schwinger--DeWitt expansion converges only for specific discrete charge values, leading to charge quantization and limitations on the existence of the evolution operator kernel.

Findings

Expansion converges for certain potentials and specific charge values.

For non-quantized charges, the expansion diverges and solutions have essential singularities.

The evolution operator kernel exists only for quantized charges and specific potentials.

Abstract

The Schwinger--DeWitt expansion for the evolution operator kernel is used to investigate analytical properties of the Schr\"odinger equation solution in time variable. It is shown, that this expansion, which is in general asymptotic, converges for a number of potentials (widely used, in particular, in one-dimensional many-body problems), and besides, the convergence takes place only for definite discrete values of the coupling constant. For other values of charge the divergent expansion determines the functions having essential singularity at origin (beyond usual $\delta$-function). This does not permit one to fulfil the initial condition. So, the function obtained from the Schr\"odinger equation cannot be the evolution operator kernel. The latter, rigorously speaking, does not exist in this case. Thus, the kernel exists only for definite potentials, and moreover, at the considered examples the charge may have only quantized values.