RATE OF CONVERGENCE OF CALCULATIONS WITH ONE-DIMENSIONAL DIRICHLET WAVE-FUNCTIONS

The convergence of numerical methods to compute the bound states of th e one-dimensional Schrodinger equation H psi = E psi in [0, infinity) by means of numerical solutions psi(Rn) of the Dirichlet eigenproblem H-R psi(R) = E(R) psi(R) in a box [0,R], is studied. It is seen that a pproximating sequences {psi(n)}(n=1)(infinity) that converge correctly to psi in the L(2) norm may have an intrinsic divergent behavior char acterized geometrically by an increasing separation between the asympt otic tails of psi(n) and psi as n --> infinity. It is shown that numer ical Dirichlet wave functions psi(Rn) obtained from standard methods c annot exhibit this divergent behavior as R, n --> infinity, and only r ounding errors may affect their convergence when R is greater than cer tain distance R(N, M(D)) that depends on the method M(D) in question, the precision machine N, and the state psi. An energy criterion to fin d R(N, M(D)) is suggested, and an estimation of the convergence rate o f expectation values from the exact Dirichlet function psi(R) as R --> infinity is given. (C) 1997 John Wiley & Sons, Inc.