Scattering at Zero Energy for Attractive Homogeneous Potentials (original) (raw)

2009, Annales Henri Poincaré

We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x| −μ with 0 < μ < 2 and γ > 0. It turns out to be the propagator for the wave equation on the sphere at time μπ 2−μ. 2 J. Dereziński and E. Skibsted Ann. Henri Poincaré see [4, Example 4.3]. Whence the deflection angle of such trajectories equals − μπ 2−μ. In particular, for attractive Coulomb potentials it equals −π, which corresponds to the well-known fact that in this case zero-energy orbits are parabolas (see [15, p. 126] for example). One can ask whether a similar behavior can be seen at the quantum level. Our analysis shows that indeed this is the case. Our main result is stated in terms of the unitary group e iθΛ generated by a certain self-adjoint operator Λ on L 2 (S d−1). The operator Λ is defined by setting ΛY = (l + d/2 − 1)Y if Y is a spherical harmonic of order l. e iθΛ can be called the propagator for the wave equation on the sphere. Note that for any θ, the distributional kernel of e iθΛ can be computed explicitly and its singularities appear at ω • ω = cos θ. This is expressed in the following fact [16]: Proposition 1.1. e iθΛ equals 1. c θ I, where I is the identity, if θ ∈ π2Z; 2. c θ P , where P is the parity operator (given by τ (ω) → τ (−ω)), if θ ∈ π(2Z+1); 3. the operator whose Schwartz kernel is of the form c θ (ω • ω − cos θ + i0) − d 2 if θ ∈]π2k, π(2k + 1)[ for some k ∈ Z; 4. the operator whose Schwartz kernel is of the form c θ (ω • ω − cos θ − i0) − d 2 if θ ∈]π(2k − 1), π2k[ for some k ∈ Z. We also remark that for all θ, the operator e iθΛ belongs to the class of Fourier integral operators of order 0 in the sense of Hörmander [11, 12]. Let us now briefly recall some points of the time-dependent scattering theory for Schrödinger operators. Set H 0 := − 1 2 Δ and H = H 0 + V (x). If the potential V (x) is short-range, following the standard formalism, we can define the usual scattering operator. In the long-range case the usual formalism does not apply. Nevertheless, one can use one of the modified formalisms, which leads to a modified scattering operator S.