Integrated density of surface states and generalized spectral shift function for a Schrodinger operator with ergodic surface potential

In this paper we study the integrated density of surface states of a Schrod inger operator with an ergodic surface potential. We also introduce another generalised function similar to the spectral shift function known in the s cattering theory. We show that these two quantities exist and coincide. We also study certain properties: their relation with the spectrum, smoothness , asymptotic behavior.