Asymptotic behavior of Jost functions near resonance points for Wigner–von Neumann type potentials

Abstract

In this work are considered radial Schrödinger operators −ψ″+V(r)ψ=Eψ, where V(r)=a sin br/r+W(r) with W(r) bounded, W(r)=O(r−2) at infinity (a,b real). The asymptotic behavior of the Jost function and the scattering matrix near the resonance point E0=b2/4 are studied. If ‖a‖>‖b‖, then this point may be an eigenvalue embedded in the continuous spectrum. The leading behavior of the Jost function for all values of a and b were determined. Somewhat surprisingly, situations were found where the Jost function becomes singular as E→E0 even if E0 is an embedded eigenvalue. Moreover, it was found that the scattering matrix is always discontinuous at E0 except in a few special cases. It is also shown that the asymptotics for the Jost function and the scattering matrix hold under weaker assumptions on W(r), in particular an angular momentum term l(l+1)r−2 may be incorporated into W(r). The results were also applied to a whole line problem with a potential V(x) such that V(x)=0 for x<0 and V(x) of Wigner–von Neumann type for x>0, and the behavior of the transmission and reflection coefficients as E→E0 was also studied.