Spectral shift via "lateral" perturbation

We consider a compact perturbation $H_0=S+K_0^{\ast }{K}_0$ 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 $K_{0}f=0$. 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 $H(K)=S + K^* K$\mathrm{.}Weshowthattheeigenvalueasafunctionof$K$ 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.