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.