Hamiltonians for Quantum Systems with Contact Interactions

TL;DR

This paper develops a new boundary condition approach to construct stable Hamiltonians for multi-particle quantum systems with contact interactions, avoiding the fall to the center instability and analyzing various limiting cases.

Contribution

It introduces a modified boundary condition for contact interactions that ensures stability and constructs physically reasonable Hamiltonians for multi-particle systems.

Findings

Avoids fall to the center instability with new boundary conditions.

Constructs Hamiltonians for bosonic and distinguishable particle systems.

Analyzes the limit of infinite mass, leading to non-local point interactions without ultraviolet issues.

Abstract

We discuss the problem of constructing self-adjoint and lower bounded Hamiltonians for a system of $n>2$ non-relativistic quantum particles in dimension three with contact (or zero-range or $\delta$) interactions. Such interactions are described by (singular) boundary conditions satisfied at the coincidence hyperplanes, i.e., when the coordinates of two particles coincide. Following the line of recent works appeared in the literature, we introduce a boundary condition slightly modified with respect to usual boundary condition one has in the one-body problem. With such new boundary condition we can show that the instability property due to the fall to the center phenomenon described by Minlos and Faddeev in 1962 is avoided. Then one obtains a physically reasonable Hamiltonian for the system. We apply the method to the case of a gas of $N$ interacting bosons and to the case of $N$ distinguishable particles of equal mass $M$ interacting with a different particle. In the latter case we also discuss the limit of the model for $M \longrightarrow +\infty$. We show that in the limit one obtains the one-body Hamiltonian for the light particle subject to $N$ (non-local) point interactions placed at fixed positions. We will verify that such non-local point interactions do not exhibit the ultraviolet pathologies that are present in the case of standard local point interactions.