Decay rates of bound states at the spectral threshold of multi-particle Schr\"odinger operators

TL;DR

This paper investigates how eigenfunctions decay at the spectral threshold for multi-particle Schrödinger operators, providing asymptotic descriptions based on system dimension and particle count.

Contribution

It offers new asymptotic decay rates for eigenfunctions at the spectral threshold in multi-particle quantum systems, especially in the critical case where the eigenvalue matches the spectrum's bottom.

Findings

Eigenfunction decay rates depend on dimension and particle number.

Asymptotic behavior characterized at the spectral threshold.

Results applicable to systems with three or more particles.

Abstract

We consider $N$-body Schr\"odinger operators with $N\geq3$ particles in dimension $d\geq 3$ in the critical case when the lowest eigenvalue coincides with the bottom of the essential spectrum of the operator. We give the asymptotic behaviour of the corresponding eigenfunction in dependence of the dimension and the number of particles of the system.