Moritz, Michael J., Eltschka, Christopher and Friedrich, Harald
(2001)
Nearthreshold quantization and level densities for potential wells with weak inversesquare tails.
Physical Review A (PRA) 64 (2), 022101.
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Other URL: http://link.aps.org/doi/10.1103/PhysRevA.64.022101
Abstract
For potential tails consisting of an inversesquare term and an additional attractive 1/r^m term, V(r)∼[ħ^2/(2M)][(γ/r^2)(β^(m2)/r^m)], we derive the nearthreshold quantization rule n=n(E) which is related to the level density via ρ=dn/dE. For a weak inversesquare term, 1/4 < γ <3/4 (and m>2), the leading contributions to n(E) are n=^(E→0) AB(E)^sqrt[γ+1/4], so ρ has a singular ...
Abstract
For potential tails consisting of an inversesquare term and an additional attractive 1/r^m term, V(r)∼[ħ^2/(2M)][(γ/r^2)(β^(m2)/r^m)], we derive the nearthreshold quantization rule n=n(E) which is related to the level density via ρ=dn/dE. For a weak inversesquare term, 1/4 < γ <3/4 (and m>2), the leading contributions to n(E) are n=^(E→0) AB(E)^sqrt[γ+1/4], so ρ has a singular contribution proportional to (E)^(sqrt[γ+1/4]1) near threshold. The constant B in the nearthreshold quantization rule also determines the strength of the leading contribution to the transmission probability through the potential tail at small positive energies. For γ=0 we recover results derived previously for potential tails falling off faster than 1/r^2. The weak inversesquare tails bridge the gap between the more strongly repulsive tails, γ≥3/4, where n(E)=^(E→0)A+O(E) and ρ remains finite at threshold, and the strongly attractive tails, γ<1/4, where n=^(E→0)Bln(E/A), which corresponds to an infinite dipole series of bound states and connects to the behavior n=^(E→0)A+B E^((1/2)(1/m)), describing infinite Rydberglike series in potentials with longerranged attractive tails falling off as 1/r^m, 0<m<2. For γ=1/4 (and m>2) we obtain n(E)=^(E→0)A+C/ln(E/B), which remains finite at threshold.
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Item type:  Article 

Date:  6 July 2001 

Institutions:  UNSPECIFIED 

Identification Number:  Value  Type 

10.1103/PhysRevA.64.022101  DOI 


Classification:  Notation  Type 

03.65.Ge, 03.65.Sq, 03.65.Xp  PACS 


Keywords:  semiclassics, WKB, bound states, scattering 

Dewey Decimal Classification:  500 Science > 530 Physics 

Status:  Published 

Refereed:  Yes, this version has been refereed 

Created at the University of Regensburg:  No 

Item ID:  8972 
