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

Abstract

Necessary and sufficient conditions and also simple sufficient conditions are given for the self-adjoint operators associated with the second-order linear differential expression τ(y)= 1 w (−(py′)′+qy) 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∫ a b ( p∣ f′∣ r + q∣ f∣ r )) 1/ r into a weighted Banach space with norm(∫ a b w∣ f∣ s ) 1/ s .