On the integrated density of surface states and a spectral shift generalized function for an ergodic surface potential

In this Note we study the integrated density of surface states of a Schrodi nger operator with an ergodic surface potential. We also introduce another generalized function similar to the spectral shift function known in the sc attering theory. We show that these two quantities exist and coincide. We a lso study certain properties: their relations with the spectrum, smoothness , asymptotic behavior. (C) 1999 Academie des Sciences/Editions scientifique s et medicales Elsevier SAS.