Spectral shift via "lateral" perturbation

We consider a compact perturbation H-0 = S + K-0* K-0 of a self-adjoint operator S with an eigenvalue lambda degrees below its essential spectrum and the corresponding eigenfunction f . The perturbation is assumed to be "along" the eigenfunction f , namely K(0)f = 0. The eigenvalue lambda degrees belongs to the spectra of both Ho and S. Let S have a more eigenvalues below lambda degrees than Ho; a is known as the spectral shift at lambda degrees. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue lambda degrees in the spectrum of H(K) = S + K* K. We show that the eigenvalue as a function of K has a critical point at K = K-0 and the Morse index of this critical point is the spectral shift a. A version of this theorem also holds for some non-positive perturbations.