An extension of Rellich’s inequality

In this paper we present a theorem which extends the results of an inequality originally due to Franz Rellich [4]. The theorem by Rellich establishes an inequality widely used in the spectral theory of partial differential operators. Our theorem allows for a broader range of application by extending the class of functions to which the theorem is applicable. Many authors call upon inequalities similar to the one established in our theorem in dealing with problems concerning essential self-adjointness of Schrodinger operators and other problems arising in oscillation theory of elliptic operators. In the first part of the paper we present Rellich's inequality and discuss some problems dealing with symmetric operators on Hilbert spaces where Rellich's inequality is a useful tool. We shall also discuss some important extensions of Rellich's work which were established by other mathematicians. One such extension was proved by W. Allegretto [1] in dealing with elliptic equations of order 2n. Another extension was established by U. W. Schmincke [5] in considering essential selfadjointness criteria of Schrodinger operators. Schmincke's extension is of particular interest to us due to his elegant proof. We follow Schmincke's method of proof. We then state and prove our generalization of Rellich's inequality along with a useful corollary. The paper concludes with a few brief comments on our result and other work which could be done with Rellich's inequality. RELLICH'S INEQUALITY Define the operator T by Tu = u cjxj4U on L 2(R n) with domain 92J(T) = CO(Rn\{O}). Then T is symmetric on 9(T). As Rellich mentions in [4] his inequality can be viewed as a means of finding the largest positive real number c such that the eigenvalues for the Friedrich's extension of T are nonnegative. Also, this could be viewed as finding the least point in the spectrum of Mu, u e CO (Rn\{O}), in the weighted Received by the editors August 5, 1987 and, in revised form, August 18, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 47F05; Secondary 26D10, 35P15, 47B25, 35J10.