On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm–Liouville problem

There is considerable interest in determining the existence of eigenvalues of the Sturm–Liouville problem−(py′)′+qy=λwy,where the independent variable x∈[0,∞) and p,q and w are real-valued functions, and λ is the spectral parameter. In general, an analytic attack on this problem is quite difficult and usually requires the use of the variational principal together with choice of suitable test functions. We show how results from functional analysis together with interval analysis and interval arithmetic can be used, not only to determine the existence of such eigenvalues, but also to compute provably correct bounds on their values.