On the convexity of the sum of the first eigenvalues of operators depending on a real parameter

  • Miriam Bareket
Brief Reports

Abstract

Letλ1 (k)≦λ2 (k)≦⋯ be the eigenvalues of an operator of a certain type depending on a real parameterk. The paper shows that under certain requirements on the operator and on the nature of its dependence onk, the sumλ1 (k)+⋯+λ N (k) is a concave function ofk, for any positive integerN.

Keywords

Mathematical Method Real Parameter Concave Function 

Zusammenfassung

Seienλ1 (k)≦2 (k)≦⋯ die Eigenwerte eines von einem reellen Parameterk abhängigen Operators. Man zeigt, daß unter gewissen Voraussetzungen über den Operator und seine Abhängigkeit vonk die Summeλ1 (k)+⋯+λN (k) für jedesN eine konkave Funktion vonk ist.

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References

  1. [1]
    S. Agmon,Lectures on elliptic boundary value problems, Van Nostrand, New York 1965.Google Scholar
  2. [2]
    M. Bareket,On the number of negative eigenvalues of a certain type of self-adjoint elliptic operators, Z. angew. Math. Phys.26, 347–355 (1975).Google Scholar
  3. [3]
    M. Bareket and B. Rulf,An eigenvalue problem related to sound propagation in elastic tubes, J. Sound and Vibration38/4, 437–449 (1975).Google Scholar
  4. [4]
    M. Bareket,Some properties of eigenvalues related to membranes and sound propagation in elastic tubes, Technical Report 80-34, Tel Aviv University, 1980.Google Scholar
  5. [5]
    R. Courant and D. Hilbert,Methods of mathematical physics (Vol. 1), Interscience, 7th printing New York 1969.Google Scholar
  6. [6]
    A. Gray and G. B. Mathews,A treatise on Bessel functions and their application to physics, 2nd ed. Dover Publ., Inc., New York 1966.Google Scholar
  7. [7]
    J. Hersch,Caractérisation variationeile d'une somme de valeurs propres consécutives; généralisation d'inégalités de Pólya-Schiffer et de Weyl, Comptes Rendus Acad. Sc. Paris252, 1714–1716 (1961).Google Scholar
  8. [8]
    G. Pólya and M. Schiffer,Convexity of functional by transplantation, J. D'Anal. Math.3, 245–345 (1953/54).Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • Miriam Bareket
    • 1
  1. 1.Div. of Applied MathematicsTel Aviv UniversityRamat-Aviv, Tel-AvivIsrael

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