Bounds on the number of scattering poles of half-Laplacian in odd dimensions, d≥ 3

2023/04/04 by Toprak, Ebru
#35P25 #47A10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.01493

Abstract

We study the scattering poles of √(-Δ) + V, where V is a compactly supported, bounded and complex valued potential. We show that the resolvent operator χR_V χ has a meromorphic continuation to the whole Riemannian surface of Λ of log z as an operator L² → L² . We then obtain the upper bound on the counting function N(r,a)= # \ z_j ∈ Λ: 0 ≤ |z_j| ≤ r, |arg z_j| ≤ a \, r >1, |a| >1 as C ⟨ a ⟩ ( ⟨ r ⟩^d + (log ⟨ a ⟩)^d) , where z_j are the poles of χR_V χ.

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