  We prove an analogue of Selberg's trace formula for a delta potential on a
hyperbolic surface of finite volume. For simplicity we restrict ourselves to
surfaces with at most one cusp, but our methods can easily be extended to any
number of cusps. In the case of a noncompact surface we derive perturbative
analogues of Maass cusp forms, residual Maass forms and nonholomorphic
Eisenstein series. The latter satisfy a functional equation as in the classical
case. We also introduce a perturbative analogue of Selberg's zeta function and
apply the trace formula to prove its meromorphic continuation to the complex
plane as well as a functional equation.
