On the Limit-Classifications of Even and Odd-Order Formally Symmetric Differential Expressions

[摘要] In this paper we consider the formally symmetric differential expression $M[cdot p]$ of any order (odd or even) ≥ 2. We characterise the dimension of the quotient space $D(T_{max})/D(T_{min})$ associated with $M[cdot p]$ in terms of the behaviour of the determinants$$detlimits_{r,sin N_n}[[f_r g_s](∞)]$$where 1 ≤ 𝑛 ≤ (order of the expression +1); here $[fg](∞) = limlimits_{x→∞}[fg](x)$, where $[fg](x)$ 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 $[fg](∞)=0$ for every $f, g in$ the maximal domain 𝛥 associated with 𝑀.

[学科分类] 数学（综合）

[关键词] Limit classification;minimal and maximal closed operators;symmetric operators;self-adjoint operators;quotient space $D(T_{max})/D(T_{min})$.