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^\circ below its essential spectrum and the corresponding eigenfunction f . The perturbation is assumed to be “along” the eigenfunction f , namely K_0f=0 . The eigenvalue \lambda^\circ belongs to the spectra of both H_0 and S . Let S have \sigma more eigenvalues below \lambda^\circ than H_0 ; \sigma is known as the spectral shift at \lambda^\circ . We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue \lambda^\circ 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 \sigma . A version of this theorem also holds for some non-positive perturbations.