Variational lower bound on the scattering length

Journal Article · Wed Oct 01 04:00:00 EDT 1975 · Phys. Rev., A, v. 12, no. 4, pp. 1297-1304

OSTI ID:4143835

Rosenberg, L; Spruch, L

The scattering length A characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on A could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on A. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on A is rigorously established provided a certin modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on A can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on A. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.

• Research Organization: Physics Department, New York University, New York, New York 10003
• NSA Number: NSA-33-004543
• OSTI ID: 4143835
• Journal Information: Phys. Rev., A, v. 12, no. 4, pp. 1297-1304, Journal Name: Phys. Rev., A, v. 12, no. 4, pp. 1297-1304; ISSN PLRAA
• Country of Publication: United States
• Language: English

Related Subjects

*ELECTRON-ATOM COLLISIONS-- SCATTERING LENGTHS
*POSITRON-ATOM COLLISIONS-- SCATTERING LENGTHS
644001*
ADIABATIC APPROXIMATION
BOUND STATE
EIGENVALUES
HAMILTONIAN FUNCTION
N60400 --Physics (Atomic & Molecular)--Collision Phenomena
N76100* --Physics (Theoretical)--General
VARIATIONAL METHODS
WAVE FUNCTIONS