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Two separation criteria for second order ordinary or partial differential operators. (English) Zbl 0937.34068
Summary: The authors generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in \(\mathbb{R}^n\). Also, for symmetric second-order ordinary differential operators they 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.

MSC:
34L05 General spectral theory of ordinary differential operators
35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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