Equidistribution of phase shifts in semiclassical potential scattering

Jesse Gell-Redman; Andrew Hassell; Steve Zelditch

doi:10.1112/jlms/jdu068

Equidistribution of phase shifts in semiclassical potential scattering

Jesse Gell-Redman, Andrew Hassell, Steve Zelditch

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Consider the Hamiltonian H:= h² Δ +V-E where Δ is the positive Laplacian on R^d, V in C₀(R^d) is a smooth, compactly supported potential, E >0 is an energy level, and h >0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix S_h(E), which lie on the unit circle S¹ C due to the unitarity of S_h(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h to 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 in S¹.

Original language	English (US)
Pages (from-to)	159-179
Number of pages	21
Journal	Journal of the London Mathematical Society
Volume	91
Issue number	1
DOIs	https://doi.org/10.1112/jlms/jdu068
State	Published - Apr 17 2015

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General Mathematics

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title = "Equidistribution of phase shifts in semiclassical potential scattering",

abstract = "Consider the Hamiltonian H:= h2 Δ +V-E where Δ is the positive Laplacian on Rd, V in C0(Rd) is a smooth, compactly supported potential, E >0 is an energy level, and h >0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix Sh(E), which lie on the unit circle S1 C due to the unitarity of Sh(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h to 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 in S1.",

author = "Jesse Gell-Redman and Andrew Hassell and Steve Zelditch",

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AU - Gell-Redman, Jesse

AU - Hassell, Andrew

AU - Zelditch, Steve

PY - 2015/4/17

Y1 - 2015/4/17

N2 - Consider the Hamiltonian H:= h2 Δ +V-E where Δ is the positive Laplacian on Rd, V in C0(Rd) is a smooth, compactly supported potential, E >0 is an energy level, and h >0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix Sh(E), which lie on the unit circle S1 C due to the unitarity of Sh(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h to 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 in S1.

AB - Consider the Hamiltonian H:= h2 Δ +V-E where Δ is the positive Laplacian on Rd, V in C0(Rd) is a smooth, compactly supported potential, E >0 is an energy level, and h >0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix Sh(E), which lie on the unit circle S1 C due to the unitarity of Sh(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h to 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 in S1.

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JF - Journal of the London Mathematical Society

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Equidistribution of phase shifts in semiclassical potential scattering

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