Spectral Shift Function in the Large Coupling Constant Limit

Given two selfadjoint operators H0 and V=V+−V−, we study the spectrum of the operator H(α)=H0+αV, α>0. We consider the quantity ξ(λ, H(α), H0), λ∈R, which coincides with Kreinʹs spectral shift function of the pair H(α), H0 if V is of trace class and study its asymptotic behavior as α→∞. Applications to differential operators are given.