Physical Review A

Scattering observables in arbitrary dimensionn≥2: An addendum

D. Bollé, Fritz Gesztesy

Published May 1, 1986• 0 citations

Abstract

Based on the theory for low-energy scattering in arbitrary dimension n\ensuremath{\ge}2, derived previously [Phys. Rev. A 30, 1279 (1984)], an analytic expansion is obtained for the two-dimensional charged scattering length around its neutral counterpart, taking into account explicitly the finite extent of the charge.

DOI: 10.1103/physreva.33.3517

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Physical Review A

Scattering observables in arbitrary dimensionn≥2: An addendum

D. Bollé, Fritz Gesztesy

Published May 1, 1986• 0 citations

Publisher

Abstract

DOI: 10.1103/physreva.33.3517

Related papers

Scattering length and effective range for charged-particle scattering in a plane and in higher dimensions

BOUDEWIJN J. VERHAAR, L.P.H. de Goey, Edgar J. D. Vredenbregt

The concepts of scattering length a and effective range ${r}_{e}$ previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter \ensuremath{\gamma}). Both a and ${r}_{e}$ have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O( ${k}^{0}$ ) and O( ${k}^{2}$ ) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,\ensuremath{\gamma}) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,\ensuremath{\gamma}) for wave number k\ensuremath{\ne}0. It is shown that these quantities have a smooth behavior for \ensuremath{\gamma}\ensuremath{\rightarrow}0 and for dimension n\ensuremath{\rightarrow}2.The concepts of scattering length a and effective range ${r}_{e}$ previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter \ensuremath{\gamma}). Both a and ${r}_{e}$ have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O(${k}^{0}$) and O(${k}^{2}$) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,\ensuremath{\gamma}) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,\ensuremath{\gamma}) for wave number k\ensuremath{\ne}0. It is shown that these quantities have a smooth behavior for \ensuremath{\gamma}\ensuremath{\rightarrow}0 and for dimension n\ensuremath{\rightarrow}2.

Physical Review ASep 1, 19855 citations

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Subhanka Mal, Kingshuk Adhikary, Dibyendu Sardar +2 more

Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.

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Ground-state energy of the interacting Bose gas in two dimensions: An explicit construction

Silas R. Beane

The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in $d$ space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of $N$ identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of $N$ . The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in $d$ space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of $N$ identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of $N$. The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.

arXiv:1002.3815Physical Review ADec 8, 201026 citations

The concepts of scattering length a and effective range

{r}_{e}

previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter \ensuremath{\gamma}). Both a and

{r}_{e}

have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O(

{k}^{0}

) and O(

{k}^{2}

) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,\ensuremath{\gamma}) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,\ensuremath{\gamma}) for wave number k\ensuremath{\ne}0. It is shown that these quantities have a smooth behavior for \ensuremath{\gamma}\ensuremath{\rightarrow}0 and for dimension n\ensuremath{\rightarrow}2.

A rigorous theory for low-energy scattering in quantum systems in

n

dimensions,

n>~2

, with general, possibly long-range, interactions is presented. It gives new insights into lowenergy phenomena; e.g., for

n=2

the controversy around the existence of scattering parameters is resolved. Furthermore, the important role of a parametrization of scattering observables in the study of spin-polarized quantum systems is illustrated. For

n=3

, an analytic expansion of the phase shift and scattering length for charged, finite-size particle scattering, around their neutral counterparts, is derived.

The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in

d

space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of

N

identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of

N

. The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.