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

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 $\gamma >{\gamma }_c^{\prime }$, with ${\gamma }_c^{\prime }$ 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 $\gamma <{\gamma }_c$.

14 pages. To appear in "Indam Quantum Meetings 22" proceedings

References in corpus (2)

• Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three
• Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

Cited by in corpus (2)

• Rigorous derivation of the Efimov effect in a simple model
• On the Weakness of Short-Range Interactions in Fermi Gases