Renormalized Contact Potential in Two Dimensions

TL;DR

This paper develops a renormalized approach for the two-dimensional attractive delta-function potential in quantum mechanics, leading to a novel eigenvalue problem involving nonlocal interactions for multiple particles.

Contribution

It introduces a cutoff-independent renormalized formulation and analyzes the N-particle eigenvalue problem, especially detailing the three-body case with a variational estimate.

Findings

Renormalized formulation avoids cutoff dependence.

Eigenvalue problem involves logarithm of a nonlocal Hamiltonian.

Provides variational estimate for three-body ground-state energy.

Abstract

We obtain for the attractive Dirac delta-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N-2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N-2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The 3-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.