Energy trajectories for the N-boson problem by the method of potential envelopes

Journal Article · Wed Oct 15 00:00:00 EDT 1980 · Phys. Rev., D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.22.2062· OSTI ID:5001538

Hall, R L

This paper concerns the ground-state energy E/sub N/ of a system of N identical bosons interacting via the attractive central pair potential V(r/sub j/i)=-V/sub 0/ f(r/sub j/i/a) and obeying nonrelativistic quantum mechanics. It is assumed that the potential shape f is decreasing and can be represented as the envelope of each of two complementary families of power-law potentials ..cap alpha..+..beta..r /sup p/ (one family is above f and the other below) for suitable fixed p=p/sub 1/ and p=p/sub 2/. If epsilon=-ma/sup 2/E/sub N//(N-1)h/sup 2/ and v=NmV/sub 0/ a/sup 2//2h/sup 2/, then it is proved that the entire collection of nonintersecting energy trajectories epsilon=F/sub N/(v), N=2,3,4,..., is bounded between the fixed curves (v,epsilon)-=(gamma( p)(s/sup 3/f'(s))/sup -1/, (v/2)(2f(s)+sf'(s))), where the curve parameter s>0, and p=p/sub 1/, p/sub 2/. Potentials, for example, with shapes f(r)=..cap alpha../sub 1//r+..cap alpha../sub 2//(r+..cap alpha../sub 3/)-..cap alpha../sub 4/lnr-..cap alpha../sub 5/sgn(q)r/sup tsq/, where ..cap alpha../sub i/> or =0 and or =1, have the ..gamma.. numbers ..gamma..(-1)=2 and ..gamma..(1)=12/..pi... The appropriate ..gamma.. numbers are provided for other classes of potential shape including perturbed harmonic oscillators, and also for problems in one spatial dimension. The method yields in effect a recipe for the way E/sub N/ depends on N and all the parameters of the pair potential.

• Research Organization: Department of Mathematics, Concordia University, Montreal, Quebec, Canada
• OSTI ID: 5001538
• Journal Information: Phys. Rev., D; (United States), Vol. 22:8
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BOSONS
MANY-BODY PROBLEM
GROUND STATES
BINDING ENERGY
COUPLING CONSTANTS
EIGENVALUES
HAMILTONIANS
HARMONIC OSCILLATORS
PARTICLE INTERACTIONS
QUANTUM MECHANICS
ELECTRONIC EQUIPMENT
ENERGY
ENERGY LEVELS
EQUIPMENT
INTERACTIONS
MATHEMATICAL OPERATORS
MECHANICS
OSCILLATORS
QUANTUM OPERATORS
645205* - High Energy Physics- Particle Interactions & Properties-Theoretical- Strong Interactions
Baryon No. = 0- (-1987)