Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

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 threebody 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.

Minlos-Faddeev regularization of zero-range interactions in the three-body problem

“…Starting from the first suggestion to modify the two-body zero-range interaction proposed by Minlos and Faddeev [1] it was declared [2][3][4] that the three-body problem becomes regularized, if the regularization parameter σ is sufficiently large to suppress the Efimov or Thomas effects, i. e., if σ exceeds the critical value σ c defined in (3.5).…”

“…In paper [1] it was suggested to introduce in the momentum-space integral equation a term containing the convolution-type operator K(k − k ′ ) depending on the relative momentum k between the interacting pair's center-of-mass and the third particle, whose asymptotic form K(ξ) → σ ξ 2 for ξ → ∞. The equivalent form of this regularization in the configurationspace representation was proposed in paper [2] and considered recently in [3,4], where an additional term was introduced to modify the boundary conditions at zero distance between the interacting particles, Clearly, the described procedure adds a kind of the three-body interaction, thus providing the regularization of Hamiltonian in the triple-collision point. As is well established, the zero total angular momentum and positive parity P (L P = 0 + ) are those quantum numbers, for which the regularization is certainly required, therefore, namely this case will be considered in this note.…”

“…Moreover, the condition σ ≥ σ c was assumed to provide an unambiguous description of the three-body problem. Recently, a proof of this conjecture was presented for three identical bosons [3] and for N identical bosons interacting with a distinct particle [4].…”

Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

“…We also believe that our approach and results can be generalized to the case of three different particles. For the case of a system made of N bosons in interaction with another particle, see [13].…”

Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions

“…It is worth to underline that the construction of a self-adjoint and bounded from below Hamiltonian for three, or more, interacting bosons with zero-range forces in dimension three is a challenging open problem in Mathematical Physics. Following a suggestion contained in [12], it has been recently studied ( [7], [11], [1]) a regularized version of the Hamiltonian for a system of three bosons (see also [6] for the case of N bosons interacting with an impurity). The main idea is to introduce a three-body repulsion that reduces to zero the strength of the contact interaction between two particles if the third particle approaches the common position of the first two.…”