The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying

We consider the linear, second-order, differential equation y"+((lambda)q(x))y=0 on [0,infinity) (*) with the boundary condition y(0) cos (alpha)+yʹ(0) sin (alpha)=0 for some (alpha) (element of)[0,(pi)). (**) We suppose that q(x) is real-valued, continuously differentiable and that q(x) (right arrow)0 as x (right arrow)infinity with q(not element of) L1[0,infinity). Our main object of study is the spectral function (rho) (alpha)[(lambda)] associated with (*) and (**). We derive a series expansion for this function, valid for (lambda)>= (lambda)0 where (lambda)0 is computable and establish a (lambda)1, also computable, such that (*) and (**) with (alpha)=0, have no points of spectral concentration for (lambda)>=(lambda)1. We illustrate our results with examples. In particular we consider the case of the Wigner –von Neumann potential.