Two Separation Criteria for Second Order Ordinary or Partial Differential Operators

We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in n . Also, for symmetric second-order ordinary differential operators we show that lim sup t→c (pq′)′/q2 = θ < 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 −∆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.