Potentials with Convergent Schwinger–DeWitt Expansion

Convergence of the Schwinger–DeWittexpansion for the evolution operator kernel for specialclass of potentials is studied. It is shown that thisexpansion, which is in the general case asymptotic,converges for the potentials considered (widely used, inparticular, in one-dimensional many-body problems), andthat convergence takes place only for definite discretevalues of the coupling constant. For other values of the charge, a divergent expansiondetermines the kernels having essential singularity atthe origin (beyond the usual δ-function). If oneconsiders only this class of potentials, then one can avoid many problems connected withasymptotic expansions, and one gets a theory withdiscrete values of the coupling constant that is incorrespondence with the discreteness of charge innature. This approach can be applied to quantum fieldtheory.