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

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 $G^{(V)}(\vec x,\vec y;E)$, ($\vec x,\vec y\in\bbbr^2,\bbbr^3$) 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.