Path Integral Discussion of Two and- Three-Dimensional $\delta$-Function Perturbations

Abstract

The incorporation of two- and three-dimensional $\delta$-function perturbations into the path-integral formalism is discussed. In contrast to the one-dimensional case, a regularization procedure is needed due to the divergence of the Green-function $G^{(V)}(\vec{x},\vec{y};E)$, ($\vec{x},\vec{y}\in {\mathbb{R}}^2,{\mathbb{R}}^3$) for $\vec{x}=\vec{y}$, corresponding to a potential problem $V(\vec{x})$. The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for $\delta$-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

32 pages, AmSTeX

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