The momentum density, n(k) of interacting many-body Fermionic systems is studied (for k>kF) using examples of several well-known two-body interaction models. It is shown that n(k) can be approximated by a zero-range model for momenta k less than about 0.1/re, where re the effective range. If the scattering length is large and one includes the effects of a fixed value of re≠0, n(k) is almost universal for momenta k up to about 2/re. However, n(k) can not be approximated by a zero-range model for momenta k greater than about 1/(are2)1/3, where a is the scattering length, and if one wishes to maintain a sum rule that relates the energy of a two component Fermi-gas to an integral involving the density. We also show that the short separation distance, s, behavior of the pair density varies as s6.The momentum density, n(k) of interacting many-body Fermionic systems is studied (for k>kF) using examples of several well-known two-body interaction models. It is shown that n(k) can be approximated by a zero-range model for momenta k less than about 0.1/re, where re the effective range. If the scattering length is large and one includes the effects of a fixed value of re≠0, n(k) is almost universal for momenta k up to about 2/re. However, n(k) can not be approximated by a zero-range model for momenta k greater than about 1/(are2)1/3, where a is the scattering length, and if one wishes to maintain a sum rule that relates the energy of a two component Fermi-gas to an integral involving the density. We also show that the short separation distance, s, behavior of the pair density varies as s6.

arXiv:1702.03370Physics Letters BJan 3, 20188 citations

Long-range scattering of two- and three-body quantum systems

Volker Enß

Journées Équations aux dérivées partiellesJan 1, 198919 citations

Scattering theory of dilated three‐body Schrödinger operators

Erik Balslev

International Journal of Quantum ChemistryOct 1, 19785 citations

General relations for quantum gases in two and three dimensions. II. Bosons and mixtures

Félix Werner, Yvan Castin

We derive exact general relations between various observables for $N$ bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body $s$ -wave scattering length $a$ . Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by $a$ , but also by a three-body parameter ${R}_{t}$ . We then find additional relations which involve the derivative of the energy with respect to ${R}_{t}$ . In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.We derive exact general relations between various observables for $N$ bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body $s$-wave scattering length $a$. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by $a$, but also by a three-body parameter ${R}_{t}$. We then find additional relations which involve the derivative of the energy with respect to ${R}_{t}$. In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.

arXiv:1210.1784Physical Review ANov 30, 2012147 citations

s-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

Aaron Farrell, Brandon P. van Zyl

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.

arXiv:1009.1918Canadian Journal of PhysicsNov 1, 20108 citations

Two-particle Schrodinger operators with point interactions

Melih Kızılkaya

Dec 1, 20200 citations

We derive exact general relations between various observables for

N

bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body

s

-wave scattering length

a

. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by

a

, but also by a three-body parameter

{R}_{t}

. We then find additional relations which involve the derivative of the energy with respect to

{R}_{t}

. In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.