Mathematical properties of one-electron wave functions

Let (-Λ + V - E)PSI = 0 in Ωsub(R) = (x element Rsup(n): |x| >= R), PSI element L2 (Ωsub(R)) where V = V1(|x|) + V2(x) and E<0, and let (-Λ + V1(|x|) - E)v(|x|) = 0 for |x| >= R' >= R sufficiently large with v>0. It is supposed that E<0 and that V tends to zero in some sense as |x| → infinity. Conditions on V are given, so that for |x| large 0 < csub(-) <= (∫ |PSI(r,ω)|2 dω)sup(1/2) / v(r) <= csub(+) < infinity where r=|x|, ω=x/|x| and dω denotes integration over the unit sphere. Also it is proved, that, given certain conditions on V2(x), PSI(r,ω)/v(r), considerd as a function in ω with r fixed, has Hoelder-continuous ksup(th) derivatives in ω. This result holds uniformly in r for r>=R'. The conditions on V(x) include e.g. Hamiltonians describing one electron in the field of fixed nuclei or the case of short range potentials, i.e. |V(x)| <= Crsup(-1-epsilon) for |x| large, epsilon>0. (Author)