Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T01:30:02.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone convergence theorems for variational triples with applications to intermediate problems

Published online by Cambridge University Press:  14 November 2011

R. D. Brown
R. D. Brown
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

The variational eigenvalue problem for a real quadratic form b with respect to a positive definite form a on a vector space V may be represented by the triple (b, a, V). Methods of intermediate problems provide approximations from below to the lower eigenvalues of (b, a, V) using monotone increasing sequences of such triples. It is shown that every such approximation method is canonically equivalent to Weinstein's method. General convergence theorems are proved for such methods. These results generalise known convergence results for increasing sequences of quadratic forms. The results are applied to some specific approximation methods and are illustrated using a differential eigenvalue problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 1-2 , 1991 , pp. 39 - 58
Copyright
Copyright © Royal Society of Edinburgh 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronszajn, N.. Approximation methods for eigenvalues of completely continuous symmetric operators. Proceedings of Symposium on Spectral Theory and Differential Equations, pp. 179–202. (Stillwater, Oklahoma: Mathematics Dept. Oklahoma Agricultural and Mechanical College, 1951)Google Scholar
2Aronszajn, N. and Brown, R. D.. Finite-dimensional perturbations of spectral problems and variational approximation methods for eigenvalue problems. Part I. Finite-dimensional perturbations. Studia Math. 36 (1970), 1–74.CrossRefGoogle Scholar
3Aronszajn, N., Brown, R. D. and Butcher, R. S.. Construction of the boundary value problems for the biharmonic operator in a rectangle. Ann. Inst. Fourier (Grenoble) 23.3 (1973), 49–89.CrossRefGoogle Scholar
4Bazley, N. W.. Lower bounds for eigenvalues with applications to the helium atom. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 850–853.CrossRefGoogle Scholar
5. Bazley, N. W. and Fox, D. W.. Lower bounds to eigenvalues using operator decompositions of the form B B. Arch. Rational Mech. Anal. 10 (1962), 352–360.CrossRefGoogle Scholar
6Bazley, N. W. and Fox, D. W.. Improvement of bounds to eigenvalues of operators of the form T T. J. Res. Nat. Bur. Standards, Sect. B 68 (1964), 173–183.CrossRefGoogle Scholar
7Bazley, N. W., Fox, D. W. and Stadter, J. T.. Upper and lower bounds for the frequencies of rectangular free plates. Z. Angew. Math. Phys. 18 (1967), 445–460.CrossRefGoogle Scholar
8Beattie, C. A.. Some Convergence Results for Intermediate Problems that Displace Essential Spectra. Applied Physics Laboratory Report 65, July, 1982 (Laurel: Johns Hopkins University).Google Scholar
9Beattie, C. A. and Greenlee, W. M.. Convergence theorems for intermediate problems. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 107–122.CrossRefGoogle Scholar
10Beattie, C. A. and Greenlee, W. M.. Corrigendum: Convergence theorems for intermediate problems. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 349–350.CrossRefGoogle Scholar
11Brown, R. D.. Perturbation Matrices for Finitely perturbed Banach Systems. University of Kansas Technical Report 27 (1977).Google Scholar
12Brown, R. D.. Convergence of approximations methods for eigenvalues of completely continuous quadratic forms. Rocky Mountain J. Math. 10 (1980), 199–215.CrossRefGoogle Scholar
13Brown, R. D.. Variational approximation methods for eigenvalues. Convergence theorems. Computational Mathematics, pp. 543–558Banach Center Publications 13, (Warsaw: 1983).Google Scholar
14Brown, R. D.. Convergence criteria for Aronszajn's method and for the Bazley-Fox method. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 91–108.CrossRefGoogle Scholar
15Fox, D. W.. Lower bounds for eigenvalues with displacement of essential spectra. SIAM J. Math. Anal. 3 (1972), 617–624.CrossRefGoogle Scholar
16Fox, D. W. and Sigillo, V. G.. Bounds for energies of radial lithium. Z. Agnew. Math. Phys. 23 (1972), 392–411.CrossRefGoogle Scholar
17Gould, S. H.. Variational Methods for Eigenvalue Problems: An Introduction to the Weinstein Method of Intermediate Problems, 2nd edn (Toronto: University of Toronto Press, 1966).CrossRefGoogle Scholar
18Greenlee, W. M.. A convergent variational method of eigenvalue approximation. Arch. Rational Mech. Anal. 81 (1983), 279–287.CrossRefGoogle Scholar
19Kato, T.. Perturbation theory for linear operators, 2nd edn (Berlin: Springer, 1976).Google Scholar
20Simon, B.. Lower semicontinuity of positive quadratic forms. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 267–273.CrossRefGoogle Scholar
21Simon, B.. A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Fund. Anal. 28 (1978), 377–385.CrossRefGoogle Scholar
22Stummel, F.. Diskrete Konvergenz lineares Operatoren I. Math. Ann. 140 (1970), 45–92.CrossRefGoogle Scholar
23Stummel, F.. Diskrete Konvergenz lineares Operatoren II. Math. Z. 120 (1971), 231–264.CrossRefGoogle Scholar
24Weidmann, J.. Monotone continuity of the spectral resolution and the eigenvalues. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980) 131–136.CrossRefGoogle Scholar
25Weinberger, H. F.. Variational Methods for Eigenvalue Problems (Philadelphia: SIAM, 1974).CrossRefGoogle Scholar
26Weinstein, A.. On a minimal problem in the theory of elasticity. J. London Math. Soc. 10 (1935), 184–192.CrossRefGoogle Scholar
27Weinstein, A. and Stenger, W.. Methods of Intermediate Problems for Eigenvalues. Theory and Ramifications (New York: Academic Press, 1972).Google Scholar