Correlations of Nearby Levels Induced by a Random Potential

We consider a Hamiltonian $H$ which is the sum of a deterministic part $H_0$ and of a random potential $V$. For finite $N \times N$ matrices, following a method introduced by Kazakov, we derive a representation of the correlation functions in terms of contour integrals over a finite number of variables. This allows one to analyse the level correlations, whereas the standard methods of random matrix theory, such as the method of orthogonal polynomials, are not available for such cases. At short distance we recover, for an arbitrary $H_0$, an oscillating behavior for the connected two-level correlation.