Vestigial effects of singular potentials in diffusion theory and quantum mechanics

Abstract

Repulsive singular potentials of the form λV (x) =λ‖x−c‖^−α, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural way to various self−adjoint extensions of the formal differential operator.