The Birman-Schwinger principle on the essential spectrum

Let H-0 and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H-0 and H corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair H-0, H. (C) 2011 Elsevier Inc. All rights reserved.