On the limit - classifications of even and odd - order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expression M[·] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient space D(T max)/D(T min) associated with M[·] in terms of the behaviour of the determinants det r,s∈N n [[ f r g s ](∞)] where 1 ≤ n ≤ (order of the expression +1); here [ f g](∞) = lim x→∞ [ f g](x), where [ f g](x) is the sesquilinear form in f and g associated with M. These results generalise the well-known theorem that M is in the limit-point case at ∞ if and only if [ f g](∞) = 0 for every f , g ∈ the maximal domain ∆ associated with M.