N-body problem with contact interactions

TL;DR

This paper introduces contact interaction Hamiltonians for N-body systems in three dimensions, showing they are limits of scaled two-body potentials and exploring their spectral properties, including bound states and Efimov effect.

Contribution

The paper formalizes contact interactions as limits of scaled two-body potentials and analyzes their spectral properties in N-body quantum systems.

Findings

Contact interactions are limits of scaled two-body potentials.

Existence of three and four-body bound states near the barycenter.

Presence of Efimov effect with infinitely many bound states.

Abstract

We introduce \emph{contact interactions hamiltonians} (self-adjointoperators defined by boundary conditions) between $N$ massive particles in $R^3$, $N \geq 3$. We prove that they are limits (in strong resolvent sense) when $ \epsilon \to 0$ of interaction through a two-body potential which scales according to $ V_\epsilon (|x|) =\epsilon ^{-3} V ( \frac {|x|}{ \epsilon} )$ where $ V(|x|) $ is an integrable function. The advantage of the formalism of contact interactions is that the results do not depend on the shape of the approximating potentials. Depending on the masses and symmetries there may be three body bound states with wave function localized near the barycenter and less localized four-body bound states. For some range of masses there is an infinity of bound states with energies which accumulate geometrically to zero (Efimov effect). For contact interactions these are all the bound states which can be present in the $N$-body system. We study of zero range hamiltonians by first compactly embedding the physical space $ L^2(R^{3 (N-1)}) $ in a larger Hilbert space of more singular functions (Krein space) Estimates are done in the auxiliary space, where strong resolvent convergence when $ \epsilon \to 0$ and spectral properties are easily established. If the hamiltonian \emph{in Krein space} is positive, coming back to $L^2 (R^{3N})$ is done using the compactness of the Krein map and the fact that for positive forms weak convergence implies strong convergence. If it is not positive we use a somewhat more sophisticated operation, $ \Gamma-$convergence [Dal]. In both cases the limit operator is unique.