The xi function

Despite the fact that spectral and inverse spectral properties of one-dimensional SchrSdinger operators H=-d2/dx2§ have been extensively studied for seventy-five years, there remain large areas where our knowledge is limited. For example, while the inverse theory for operators on L2( ( -co , co)) is well understood in case V is periodic [12], [24], [25], [35], [39]-[42], [49], it is not understood in case limjx]-.o~ V(x)=-co and H has discrete spectrum. Our goal here is to introduce a special function ~(x,A) on R • associated to H which we believe will be a valuable tool in the spectral and inverse spectral theory. In a sense we will make precise, it complements the Weyl m-functions, m+ (x, A). A main application of ~ which we will make here concerns a generalization of the trace formula for SchrSdinger operators to general V's. Recall the well-known trace formula for periodic potentials: Let V(x)=V(x-t-1). Then, by Floquet theory (see, e.g., [10], [37], [44]),