Geometry and analysis in many-body scattering

Inverse problems in N-body scattering

We survey finite energy inverse results in N-body scattering, and we also sketch the proof of the extension of our recent two-cluster to two-cluster three-body result to the many-body case: this requires only minor modifications. We also indicate how to extend a free-to-free inverse result in three-body scattering to the many-body case.

Low energy inverse problems in three-body scattering

We consider low energy inverse problems in three-body scattering and show that if all unknown interactions are small in an appropriate sense then the 2-cluster to 2-cluster S-matrices given at low energies determine the Fourier transform of the effective interaction in a ball.

Exponential decay of eigenfunctions in many-body type scattering with second order perturbations

We show the exponential decay of eigenfunctions of second-order geometric many-body type Hamiltonians at non-threshold energies. Moreover, in the case of first order and small second order perturbations we show that there are no eigenfunctions with positive energy.

Scattering Theory on SL(3)/SO(3): Connections with Quantum 3-Body Scattering

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\SO(3,\RR)$ is very closely related to quantum three-body scattering. In particular, we adapt geometric constructions from recent advances in that field by one of us (A.V.), as well as from our previous work [Mazzeo-Vasy:Resolvents] concerning resolvents for product spaces, to giv…

N体时间衰减谐振子的逆散射

Inverse scattering for $N$-body time-decaying harmonic oscillators

本研究将Ishida（2025）的二体量子系统逆散射结果扩展到N体情况，证明高能限下散射算子能唯一确定所有成对相互作用势能。

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

Let H=Δ+\sum_{#a=2} V_a be a 3-body Hamiltonian, H_a the subsystem Hamiltonians, Δthe positive Laplacian of the Euclidean metric on X_0=R^n, 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(H_0)-\sum_{#a=2}(ϕ(H_a)-ϕ(H_0)) 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 addit…