Polynomials and the limit point condition

Abstract:

An $n$th order, possibly nonselfadjoint, ordinary differential expression $L$ is said to be in the limit point condition if the maximal operator ${L_M}$ in ${L_2}[0,\infty )$ is an $n$-dimensional extension of the minimal operator ${L_0}$. If range ${L_0}$ is closed, this definition is equivalent to the assertion that nullity ${L_M} + \text {nullity} {({L^ + })_M} = n$, where ${L^ + }$ is the formal adjoint of $L$. It also implies that any operator $T$ such that ${L_0} \subseteq T \subseteq {L_M}$ is the restriction of ${L_M}$ to a set of functions described by a boundary condition at zero. In this paper, we discuss the question of when differential expressions involving complex polynomials in selfadjoint expressions are in the limit point condition.