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From short-range to contact interactions in the 1d Bose gas

Marcel Griesemer, Michael Hofacker, Ulrich Linden

Published Jan 1, 2020• 12 citations
Publisher

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
10.18419/opus-13232

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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.
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.
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.
In this paper, we study the Hamiltonian for a three-dimensional Bose gas of [Formula: see text] spinless particles interacting via zero-range (also known as contact) interactions. Such interactions are encoded by (singular) boundary conditions imposed on the coincidence hyperplanes, i.e. when the coordinates of two particles coincide. It is well known that imposing the same kind of boundary conditions as in the two-body problem with a point interaction leads to a Hamiltonian unbounded from below (and thus unstable). This is due to the fact that the interaction becomes overly strong and attractive when the coordinates of three or more particles coincide. In order to avoid such instability, we develop a suggestion originally formulated by Minlos and Faddeev in 1962, introducing slightly modified boundary conditions that weaken the strength of the interaction between two particles i and j in two scenarios: (a) a third particle approaches the common position of i and j; (b) another distinct pair of particles approach each other. In all other cases, the usual boundary condition is restored. Using a quadratic form approach, we construct a class of Hamiltonians characterized by such modified boundary conditions, that are self-adjoint and bounded from below. We also compare our approach with the one developed years ago by Albeverio, Høegh-Krohn and Streit using the theory of Dirichlet forms [J. Math. Phys. 18 (1977) 907–917]. In particular, we show that the N-body Hamiltonian defined by Albeverio et al. is a special case of our class of Hamiltonians. Furthermore, we also introduce a Dirichlet form by considering a more general weight function, and we prove that the corresponding N-body Hamiltonians essentially coincide with those constructed via our method.