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Lower bounds for eigenvalues of self-adjoint problems

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Publication Date
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PMC
Keywords
  • Physical Sciences: Mathematics
Disciplines
  • Mathematics

Abstract

The equation y″ + [λ - q(x)]y = 0 on (0, ∞) or (-∞, ∞), in which q(x) → ∞ as x → ∞ or x → ± ∞, has a complete set of eigenfunctions with discrete eigenvalues {λn}n=0∞. We derive an inequality that contains λn, by using a quick and elementary method that does not employ a comparison theorem or assume anything special. Explicit lower bounds for λn can often be easily obtained, and three examples are given. The method also gives respectable lower bounds for λn in the classical Sturm—Liouville case.

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