Finite-dimensional approximation of the differential operator in problems of quantum mechanics (Q1073555)

This paper considers an approximation, \(\hat T,\) to a differential operator T which coincides with T on some finite set of functions \(\chi_ i\) so that \(\hat TT^{-1}\chi_ i=\chi_ i\) \(i=1,...,N\). It is applied to a general Schrödinger equation where T denotes the exact kinetic-energy operator. In order for appropriate functions \(\chi_ i\) to be found the minima of functionals such as \(F_ n=| <\psi | (T-\hat T)^ n\psi >|\) are used where \(\psi\) is the solution of the approximated Schrödinger equation \((\hat T+V-E)\psi =0.\) More complicated functionals are also used and these procedures determine good functions \(\chi_ i\) and the solution \(\psi\) in terms of these functions. Examples are presented from simple model systems showing in one case that the Born approximation is a special case of this theory when one function is used. The example \((d^ 2/dx^ 2+(e^{-\mu x}/x)+E)\psi =0\) is used more extensively as an eigenvalue problem and as a scattering theory problem.