Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions

TL;DR

This paper analyzes a regularized Hamiltonian model for three bosons with contact interactions, establishing conditions for its stability and self-adjointness, and demonstrating the optimality of the regularization threshold.

Contribution

It provides a rigorous construction of the Hamiltonian for three bosons with zero-range forces using Minlos-Faddeev regularization and identifies the optimal threshold for stability.

Findings

Hamiltonian is self-adjoint and bounded below for g > g_c

Simplified proof of stability for g > g'_c

Threshold g_c is proven to be optimal

Abstract

We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider the so-called Minlos-Faddeev regularization of such Hamiltonian, heuristically corresponding to the introduction of a three-body repulsion. We review the main concerning results recently obtained. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $\mathcal H$ can be constructed provided that the strength $\gamma$ of the three-body force is larger than a threshold parameter $\gamma_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $\gamma > \gamma'_c$, with $\gamma'_c$ strictly larger than $\gamma_c$. Finally, we show that the threshold value $\gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $\gamma<\gamma_c$.