Zero range (contact) interactions conspire to produce Efimov trimers and quadrimers

TL;DR

This paper introduces contact (zero range) interactions as a class of self-adjoint extensions of the N-body Schrödinger Hamiltonian, revealing their role in producing Efimov trimers and quadrimers and analyzing their spectral properties.

Contribution

It defines contact interactions via boundary conditions, analyzes their spectral properties, and proves their limit behavior as short-range potentials, extending understanding of N-body quantum systems.

Findings

Efimov spectrum arises from contact interactions of two pairs.

Contact interactions are limits of short-range potentials as epsilon approaches zero.

Spectral properties depend on masses and statistics of particles.

Abstract

We introduce \emph{contact (zero range) interactions } , a special class of self-adjoint extensions of the N-body Schr\"odinger free hamiltonian $ H_0$ restricted to functions with support away from the \emph{contact manifold} $ \Gamma \equiv \cup \Gamma_{i,j} \;\;\; \Gamma_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3 $. These extensions are defined by boundary conditions at $ \Gamma$. We discuss the spectral properties as function of the masses and the statistics The (Efimov) spectrum is entirely due "conspiracy" of the contact interactions of two pairs. These states are called in Theoretical Physics \emph{trimers} if the two pairs have an element in common, \emph{quadrimers} otherwise. The analysis can be extended to the case in which there is a regular two-body potential, but then the spectral properties cannot be given explicitly. We prove that these interactions are limit (in strong resolvent sense) when $ \epsilon \to 0$ of N-body hamiltonians $ H_\epsilon = H_0+ \sum_{i,j} \frac {1}{\epsilon^3}V_{i,j}( \frac {|x_i-x_j|}{\epsilon}) ,\; V_{i,j} \in L^1 $. The result is \emph{ independent of the shape of the potential} Strong resolvent convergence implies convergence of spectra.This makes contact interaction a valuable tool in the study of the spectrum of a system of $N$ particles interacting through potentials of very short range.