Open AccessLower bounds for eigenvalues of self-adjoint problems
Author(s) -
Gary G. Gundersen
Publication year - 1979
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.76.11.5424
Subject(s) - eigenfunction , eigenvalues and eigenvectors , mathematics , self adjoint operator , set (abstract data type) , sturm–liouville theory , combinatorics , upper and lower bounds , pure mathematics , mathematical analysis , physics , boundary value problem , computer science , quantum mechanics , hilbert space , programming language
The equationy ″ + [λ -q (x )]y = 0 on (0, ∞) or (-∞, ∞), in whichq (x ) → ∞ asx → ∞ orx → ± ∞, 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.