Spectral shift via “lateral” perturbation

We consider a compact perturbation $H_0 = S + K_0^* K_0$ of self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The is assumed to be "along" $f$, namely $K_0f=0$. belongs spectra both $H_0$ $S$. Let have $\sigma$ more eigenvalues than $H_0$; known as spectral shift at $\lambda^\circ$. now allow vary in suitable space study continuation $H(K)=S K^* K$. show that function $K$ has critical point $K=K_0$ Morse index this $\sigma$. A version theorem also holds for some non-positive perturbations.