Smoothness and high energy asymptotics of the spectral shift function in many-body scattering

2001/06/25 by András Vasy, Andras Vasy, Vasy, Andras +3
Materials Science · Mathematics · Physics and Astronomy · #35P25 (Primary) 47A40 #81U10 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Lanthanide and Transition Metal Complexes #Mathematical Physics (math-ph) #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #math.SP #msc:35P25 #msc:47A40 #msc:81U10

paper · pdf · doi:10.48550/arxiv.math/0106209

30 pages, no figures

arxiv created 2001/06/25 · openalex publication_date 2001/06/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H=Δ+∑_#a=2 V_a be a 3-body Hamiltonian, H_a the subsystem Hamiltonians, Δthe positive Laplacian of the Euclidean metric on X₀=Rⁿ, V_a real-valued. Buslaev and Merkurev have shown that, if the pair potentials decay sufficiently fast, for ϕsmooth and compactly supported, the operator ϕ(H)-ϕ(H₀)-∑_#a=2(ϕ(H_a)-ϕ(H₀)) is trace class. Hence, one can define a modified spectral shift function σ, as a distribution on R, by taking its trace. In this paper we show that if V_a are Schwartz, then σis in fact smooth away from the thresholds, and obtain its high energy asymptotics. In addition, we generalize this result to N-body scattering, N arbitrary.

Citations

• Singularities of boundary value problems. I

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