Journal article

Approximation and Equidistribution of Phase Shifts: Spherical Symmetry

Kiril Datchev, Jesse Gell-Redman, Andrew Hassell, Peter Humphries



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 ..

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Awarded by Australian Research Council

Awarded by Direct For Mathematical & Physical Scien; Division Of Mathematical Sciences

Funding Acknowledgements

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.