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From Short-Range to Contact Interactions in Two-dimensional Many-Body Systems

Marcel Griesemer, Michael Hofacker

Published Mar 11, 2021• 3 citations
OpenAlex

Abstract

Quantum systems composed of $N$ distinct particles in $\R^2$ with two-body contact interactions of TMS type are shown to arise as limits - in the norm resolvent sense - of Schrodinger operators with suitably rescaled potentials.

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Quantum systems composed of NNN distinct particles in R2\R^2R2 with two-body contact interactions of TMS type are shown to arise as limits - in the norm resolvent sense - of Schrodinger operators with suitably rescaled potentials.
Abstract Quantum systems composed of N distinct particles in R2\mathbb {R}^2R2 R 2 with two-body contact interactions of TMS type are shown to arise as limits—in the norm resolvent sense—of Schrödinger operators with suitably rescaled pair potentials.
We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.
We reconsider the norm resolvent limit of −Δ+Vℓ-Δ+ V_\ell−Δ+Vℓ​ with VℓV_\ellVℓ​ tending to a point interaction in three dimensions. We are mainly interested in potentials VℓV_\ellVℓ​ modelling short range interactions of cold atomic gases. In order to ensure stability the interaction VℓV_\ellVℓ​ is required to have a strong repulsive core, such that lim⁡ℓ→0∫Vℓ>0\lim_{\ell \to 0} \int V_\ell >0limℓ→0​∫Vℓ​>0. This situation is not covered in the previous literature.
Abstract We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan–Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant.