Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrodinger operators

For the general one dimensional Schrodinger operator -d(2)/dx(2) + q(x) with real q we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-HoldenSimon-Zhao. We show that the condition q is an element of L-1 ( R) guarantees the existence of the trace formulas of order one only with certain resolvent regularizations of the integrals involved. Our principle results are simple necessary and sufficient conditions on absolute summability of the formulas under consideration. These conditions are expressed in terms of Fourier transforms related to q.