Approximation and Equidistribution of Phase Shifts: Spherical Symmetry
Kiril Datchev, Jesse Gell-Redman, Andrew Hassell, Peter Humphries
COMMUNICATIONS IN MATHEMATICAL PHYSICS | SPRINGER | Published : 2014
Consider a semiclassical Hamiltonian HV,h := h2 Δ + V − E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on Rd , V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix Sh(E) is a unitary operator on L2(Sd−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 Sh(E) can be divided into two classes: a finite number ∼ cd (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 ..View full abstract
Awarded by Australian Research Council
Awarded by Direct For Mathematical & Physical Scien; Division Of Mathematical Sciences
K. Datchev was partially supported by an NSF postdoctoral fellowship. The authors would like to thank Hamid Hezari, Volker Schlue and Steve Zelditch for very helpful conversations. A. Hassell acknowledges the support of the Australian Research Council through a Future Fellowship FT0990895 and Discovery Grant DP1095448.