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arXiv:1908.05705 · Mathematical Physics Analysis and Geometry

From Short-Range to Contact Interactions in the 1d Bose Gas

Marcel Griesemer, Michael Hofacker, Ulrich Linden

Published May 15, 2020• 11 citations
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Abstract

Abstract For a system of N bosons in one space dimension with two-body δ-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.
DOI
10.1007/s11040-020-09344-4

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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.
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.
Abstract We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.
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