[Paper Review] Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions

Simon Becker, Alessandro Michelangeli+1 more|arXiv (Cornell University)|Dec 29, 2017

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper analyzes the spectral properties of a 2+1 fermionic trimer with zero-range interactions, proving the finiteness of the discrete spectrum and monotonicity of eigenvalues with respect to the mass parameter. It identifies the essential spectrum, characterizes eigenfunction symmetries, and demonstrates the existence of bound states in physically relevant mass regimes.

ABSTRACT

We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.

Motivation & Objective

• To characterize the full spectrum of the Hamiltonian for three-point-particle systems with zero-range interactions.
• To determine the essential spectrum and discrete spectrum structure in a 2+1 fermionic system with arbitrary mass ratios above the stability threshold.
• To establish the angular symmetry of eigenfunctions and the monotonicity of eigenvalues with respect to the mass parameter.
• To demonstrate the existence of bound states in physically relevant mass regimes.

Proposed method

• Analytical treatment of the Hamiltonian with point interactions in three dimensions using functional analytic methods.
• Application of spectral theory to identify the essential spectrum via asymptotic completeness and scattering theory.
• Use of variational and perturbative techniques to localize and prove finiteness of the discrete spectrum.
• Classification of eigenfunctions by their angular momentum quantum numbers based on symmetry considerations.
• Derivation of monotonicity properties of eigenvalues using variational principles and dependence on the mass parameter.
• Construction of rigorous bounds and existence proofs for bound states in the physically relevant mass range.

Experimental results

Research questions

• RQ1 What is the structure of the essential and discrete spectrum of the 2+1 fermionic trimer with zero-range interactions?
• RQ2 How do the eigenvalues of the system depend on the mass ratio between the third particle and the identical fermions?
• RQ3 What is the angular symmetry of the eigenfunctions corresponding to bound states?
• RQ4 Under what conditions does the system support bound states in the physically relevant mass regime?
• RQ5 Is the discrete spectrum finite for arbitrary interaction strength and mass ratios above the stability threshold?

Key findings

• The essential spectrum is fully characterized and corresponds to the continuous part of the energy spectrum arising from scattering states.
• The discrete spectrum is proven to be finite for all interaction strengths and mass ratios above the stability threshold.
• Eigenfunctions exhibit definite angular momentum quantum numbers, with symmetry classification dependent on the system's exchange statistics.
• Eigenvalues are strictly monotonic with respect to the mass parameter, increasing as the third particle becomes heavier.
• Bound states exist in the physically relevant regime where the third particle is heavier than the fermions, confirming stability in this range.
• The analysis confirms the absence of infinitely many bound states, even for strong interactions, due to the finiteness of the discrete spectrum.