Journal article

The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

Jesse Gell-Redman, Andrew Hassell

International Mathematics Research Notices | OXFORD UNIV PRESS | Published : 2020

Abstract

This is the 3rd paper in a series [6, 9] 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 Sh, at some fixed energy E, for semiclassical Schrödinger operators on Rd that are perturbations of the free Hamiltonian h2 on L2(Rd) by a potential V with polynomial decay. Our assumption is that V(x) ∼ |x| −αv(xˆ) as x → ∞, xˆ = x/|x|, for some α > d, with corresponding derivative estimates. In the semiclassical limit h → 0, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in h, tends to a measure μ on S1. Moreove..

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Grants

Awarded by Australian Research Council


Funding Acknowledgements

This work was supported by the Australian Research Council through Discovery Project [DP150102419 to A. H.].