Semi-Classical and High Energy Asymptotics of the Scattering Phase for Perturbations of Elliptic Operators

Abstract

For usual Schrödinger operators the scattering phase can be defined as follows: Let us consider the free Hamiltonian ${H_0} = - \frac{{{\hbar ^2}}} {{2m}}\Delta$ and a smooth perturbation H = H ₀ + V of H ₀ such that V(x) = O(|x| ^-ρ) as |x| → + ∞, x ∈Rⁿ, the configuration space. If ρ > n then it is well known that the scattering matrix S(λ), λ > 0, for the system (H, H ₀) is a trace-class perturbation of the identity on L ²(S^{n - 1}). The scattering phase: s(λ), is then defined by the equality:

$$ detS(\lambda ) = \exp \left( { - 2i\pi s\left( \lambda \right)} \right) $$

((1))

s(λ) and its derivatives have many interesting properties as well as from physical and mathematical point of view. This notion was first introduced for central potentials. In that case S(λ) is diagonalized on the spherical harmonics: S(λ)f = (exp (2iδ _ℓ(λ))f ∀f ∈ ϰ_ℓ;(spherical harmonics of degree ℓ) [11]. We have clearly:

$$ s\left( \lambda \right) = - \frac{1} {\pi }\sum\nolimits_{\ell \geqslant 0} {\left( {2\ell + 1} \right)} {\delta _\ell }\left( \lambda \right) $$

.