Limitations of the method of complex basis functions

Abstract

The method of complex basis functions proposed by Rescigno and Reinhardt is applied to the calculation of the amplitude in a model problem which can be treated analytically. It is found for an important class of potentials, including some of infinite range and also the square well, that the method does not provide a converging sequence of approximations. However, in some cases, approximations of relatively low order might be close to the correct result. The method is also applied to 8-wave e-H elastic scattering above the ionization threshold, and spurious "convergence" to the wrong result is found.A procedure which might overcome the difficulties of the method is proposed.

Electron-impact ionization of atomic hydrogen

“…Rodwhich is precisely the case that Baumel, Crocker, and NuttaU [13] showed to be divergent. We will now give the specific form of the transformation we used to implement smooth exterior scaling.…”

“…Calculations on electron-hydrogen scattering for the case where only a few channels are energetically allowed were carried out by Burke et al [13] using the "closecoupling" method. This method uses a physically motivated expansion of the scattered wave function in terms of products of bound states and free-particle functions.…”

“…Specifically, what is required is lim=-O(@olV(E -H + is) -1 V] @o), where @ois a continuum function. Unfortunately, with ordinary complex scaling, these so-called "free-free" elements only converge for exponentially bounded potentirds V [2,13]. Our main purpose here will be to show how such a construction can be made to work even in the case of a Harniltonian with long-range interactions.…”

“…Unfortunately, as Baumel, Crocker, and Nuttall [13] have pointed out, V(r) must be exponentially bounded for Eq. (29) to converge since sin(kr) diverges exponentially under coordinate rotation.…”

“…A solution of the full scattering problem requires matrix elements of the resolvent between continuum functions. Unfortunately, the method of complex scaling as originally presented only provides convergent expressions for these quantities in the case of interaction potentials that fall off exponentially [2,13], which would appear to exclude most of the problems encountered in atomic and molecular physics. Although methods based on complex scaliig or, more accurately, on the use of complex basis functions [8] have been proposed to tackle this harder problem, it is probably fair to say that, after many years, no definitive method for entirely circumventing the specification of boundary conditions has emerged.…”