A Many-body Problem with Point Interactions on Two Dimensional Manifolds

Abstract

A non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the $\beta$ function is exactly calculated for the general case, which includes all particle numbers.

28 pages; typos are corrected, three figures are added

References in corpus (11)

• Aharonov-Bohm Scattering, Contact Interactions and Scale Invariance
• Perturbative Renormalization in Quantum Mechanics
• Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. I. General Theory
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• Point interactions in a strip
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• Point Interaction in two and three dimensional Riemannian Manifolds
• Renormalized Contact Potential in Two Dimensions
• Nonrelativistic Lee model in three dimensional Riemannian manifolds
• Non-relativistic Lee Model on two Dimensional Riemannian Manifolds
• Non-commutative field theory approach to two-dimensional vortex liquid system

Cited by in corpus (5)

• One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
• From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems
• Nondegeneracy of the Ground State for Nonrelativistic Lee Model
• Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications
• Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution