Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems

Necessary and sufficient conditions and also simple sufficient conditions are given for the self-adjoint operators associated with the second-order linear differential expression on [a,b) to have discrete spectrum. Here the coefficients of τ are non-negative and satisfy minimal smoothness conditions. These results follow from compact embedding theorems from a weighted one-dimensional Sobolev space with norm∫ab(p f′ r+q f r))1/r into a weighted Banach space with norm(∫abw f s)1/s .