  A quantum field described by the field operator $\Delta_{a}=\Delta+
a\delta_\Sigma$ involving a $\delta$-like potential is considered.
Mathematically, the treatment of the $\delta$-potential is based on the theory
of self-adjoint extension of the unperturbed operator $\Delta$. We give the
general expressions for the resolvent and the heat kernel of the perturbed
operator $\Delta_{a}$. The main attention is payed to $d=2$ $\delta$-potential
though $d=1$ and $d=3$ cases are considered in some detail. We calculate
exactly the heat kernel, Green's functions and the effective action for the
operator $\Delta_{a}$ in diverse dimensions and for various spaces $\Sigma$.
The renormalization phenomenon for the coupling constant $a$ of $d=2$ and $d=3$
$\delta$-potentials is observed. We find the non-perturbative behavior of the
effective action with respect to the renormalized coupling $a_{ren}$.
