Bounds for the points of spectral concentration of one-dimensional Schrodinger operators

We investigate the phenomenon of spectral concentration for one-dimensional Schrodinger operators with decaying potentials on the half-line. For suitable classes of short range and long range potentials, we outline systematic procedures which enable numerical estimates of upper bounds for points of spectral concentration to be obtained. Our approach involves use of the Riccati equation to construct appropriate convergent series for a generalised Dirichlet m-function, from which the existence and properties of derivatives of the corresponding spectral functions can be established. An incidental outcome in the case of long range potentials is that upper bounds for embedded singular spectrum can also be obtained.