Asymptotic-behavior of Jost functions near resonance points for Wigner-Vonneumann type potentials

Abstract

In this work are considered radial Schrodinger operators - psi" + V(r)-psi = E-psi, 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 E(o) = 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-->E(o) even if E(o) is an embedded eigenvalue. Moreover, it is found that the scattering matrix is always discontinuous at E(o) 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 are 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-->E(o) is also studied.