2015/09/11 by Jesse Gell‐Redman, Jesse Gell-Redman, Gell-Redman, Jesse +2 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.48550/arxiv.1509.03468
35 pages
arxiv created 2015/09/11 · openalex publication_date 2015/09/11 · arxiv updated 2015/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the third paper in a series 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(E), for semiclassical Schrödinger operators on ℝd which are perturbations of the free Hamiltonian by a potential V with polynomial decay. Our assumption is that V(x) ∼ |x|-α v( x) as 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 \mathbbS1. Moreover, μ is the pushforward from ℝ to ℝ / 2 πℤ = \mathbbS1 of a homogeneous distribution ν of order β depending on the dimension d and the rate of decay α of the potential function. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of \mathbbS1. The proof relies on an extension of results of the second author and Wunsch on the classical Hamiltonian dynamics and semiclassical Poisson operator to the class of potentials under consideration here.