MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 273-292 (1999)

Two separation criteria for second order ordinary or partial differential operators

R. C. Brown, D. B. Hinton

R. C. Brown, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A., e-mail:; D. B. Hinton, Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A., e-mail:

Abstract: 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.

Keywords: separation, ordinary or partial differential operator, limit-point, essentially self-adjoint

Classification (MSC2000): 34L05, 35P05, 47F05, 34L40, 26D10

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