Lower bounds for eigenvalues of self-adjoint problems.
The equation y'' + [lambda - q(x)]y = 0 on (0, infinity) or (-infinity, infinity), in which q(x) --> infinity as x --> infinity or x --> +/- infinity, has a complete set of eigenfunctions with discrete eigenvalues {lambda(n)}(n=0) (infinity). We derive an inequality that contains lambda(n), by using a quick and elementary method that does not employ a comparison theorem or assume anything special. Explicit lower bounds for lambda(n) can often be easily obtained, and three examples are given. The method also gives respectable lower bounds for lambda(n) in the classical Sturm-Liouville case.
Gundersen
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Mathematics Department, University of New Orleans, New Orleans, Louisiana 70122.