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Physical Review A

Adiabatic optical potential and variational lower bounds on scattering parameters near threshold

Yukap Hahn

Published Dec 1, 1975• 6 citations
Publisher

Abstract

The exact elastic optical potential is compared with an analogous potential in the adiabatic approximation, which is shown to give a lower bound on the scattering length. A variational-bound procedure is formulated to estimate the scattering length and also the scattering phase shift near the elastic threshold.
DOI
10.1103/physreva.12.2402

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The scattering length AAA characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on AAA 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 AAA. 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 AAA is rigorously established provided a certain 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 AAA 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 AAA. 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.
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We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.