The Birman–Schwinger principle on the essential spectrum

Let H0 and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H0 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 H0, H.