On the Shrödinger operator with homogeneous potential of degree zero

In this talk we are concerned with the Schrödinger operator $H=-\Delta+W(x)+V(x)$, $V$ and $W$ are real valued. The potential W, is homogeneous of degree zero.

First we describe some spectral properties of the non-perturbed Hamiltonian $H_0=-\Delta+W(x)$.  Second, we explain why the set of critical  points of the potential W  should be important for scattering theory. In the second part, we study the effect of the decaying perturbation V. In particular,  at high energy, a full complete asymptotic expansion of the spectral shift function is given.
Finally, we study the distribution of eigenvalues of  the operator $H_\mu=-\Delta+V+\mu W$ when  $\mu$ tends to infinity.