Approximation and Equidistribution of Phase Shifts: Spherical Symmetry

Abstract

Abstract. Consider a semiclassical HamiltonianH V,h := h 2 ∆ + V − E where h > 0 is a semiclassical parameter, ∆ is the positive Laplacian on R d , V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix S h (E) is a unitary operator on L 2 (S d−1 ), hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at 1.We show under certain additional assumptions on the potential that the eigenvalues of S h (E) can be divided into two classes: a finite number ∼ c d (R √ E/h) d−1 , as h → 0, where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to 1. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively.A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius R.

Equidistribution of phase shifts in semiclassical potential scattering

“…The existence of potentials satisfying Assumption 2.2 was established in [6]. Indeed, Assumption 2.2 is weaker than the dynamical assumption made in [6], and example of potentials satisfying the strong assumption of that paper are established therein. See Section 6 for details.…”

The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

“…When V ∼ c/r α , it is straightforward to compute that r m = η(1 + O(η −α )) and Σ(η) = O(η −α ). This is the third paper in a series analyzing the asymptotic distribution of the phase shifts in the semiclassical limit using geometric microlocal techniques, the first two works of which consider smooth compactly supported potentials V [5,8]. It is instructive to compare the Main Theorem with the main result of [8], which is where I is the set of (ω, η) ∈ T * S d−1 associated to bicharacteristics that meet the support of V .…”

“…Indeed, in this case the canonical relation of S h is the graph of the map of T * S 1 −→ T * S 1 taking a point (ω, η) to (ω + Σ(η), η), where (ω, η) corresponds to a straight ray x 0 (t) = ωt + η and η ⊥ ω. Here, the scattering angle is given explicitly by the formula [5,Eqn. 2.6]…”

“…In[12], the sojourn relation was viewed as a Legendre submanifold of a space with one extra dimension, with the extra coordinate being the variable denoted φ below. Here, we take the view that φ is a function defined on SR 5. Our partial compactification serves to make the energy surface {|ξ| 2 + V = E} compact, which is all that matters.…”

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This is the third paper in a series [5,8] analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix, S h (E), for semiclassical Schrödinger operators on R d which are perturbations of the free Hamiltonian by a potential V with polynomial decay.

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