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Annali di Matematica Pura ed Applicata

, Volume 77, Issue 1, pp 317–326 | Cite as

A method for determining upper bounds for the smallest eigenvalue in a class of singular problems

  • Walter Leighton
Article

Summary

A method employing the calculus of variations for determining upper bounds of the smallest eigenvalue in a class of singular, second-order, linear problems is provided. Applications to the equation
$$y'' + 2\alpha [(x - \alpha )y]' + 4\beta y(1 - x^2 )^{ - 1} = 0$$
and to Bessel's equation illustrate the procedure.

Keywords

Linear Problem Small Eigenvalue Singular Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Walter Leighton,Principal quadratic functionals, Trans. of the Amer. Math. Soc., Vol. 67, No. 2 (1949), pp. 253–274.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    Walter Leighton andA. D. Martin,Quadratic functionals with a singular end point, Trans. of the Amer. Math. Soc., Vol. 78 (1955), pp. 98–128.CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Walter Leighton,Comparison theorems for linear differential equations of second order, Proc. of the Amer. Math. Soc., Vol. 13 (1962), pp. 603–610.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    G. F. Miller.The evaluation of eigenvalues of a differential equation arising in a problem in genetics, Proc. of the Cambridge Phil. Soc., Vol. 58 (1962), pp. 588–593.MATHCrossRefGoogle Scholar
  5. [5]
    Marston Morse andWalter Leighton,Singular quadratic functionals, Trans. of the Amer. Math. Soc, Vol. 40, No. 2 (1936), pp. 252–286.CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    G. Sansone andL. Merli,L'equazione y′' + 2α[(x − a)y]′ + 4βy(1 − x 2)−1= 0, Annali di Matematica, Serie Quarta, LXVII (1965), pp. 95–112.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Edmond Tomastik,Singular quadratic functions of n dependent variables, Trans. Amer. Math. Soc., Vol. 124, No. 1 (1966), pp. 60–76.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    E. Jahnke andF. Emde,Tables of functions with formulae and curves, Dover, New York (1943).Google Scholar

Copyright information

© Nicola Zanichelli Editore 1967

Authors and Affiliations

  • Walter Leighton
    • 1
  1. 1.Columbia

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