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separation; ordinary or partial differential operator; limit-point; essentially selfadjoint
We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in $\Bbb R^n$. Also, for symmetric second-order ordinary differential operators we show that $\limsup_{t\to c} (pq')'/q^2=\theta<2$ where $c$ is a singular point guarantees separation of $-(py')'+qy$ on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that $-\Delta y+qy$ is separated on its minimal domain if $q$ is superharmonic. For $n=1$ the criterion is used to give examples of a separation inequality holding on the domain of the minimal operator in the limit-circle case.
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