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How rare are multiple eigenvalues? (English) Zbl 0942.47012

Introducing the new concept of codimension in infinite- dimensional space the author considers a differentiable family \(A(q)\) of self-adjoint operators on a Hilbert space \(H\), indexed by a parameter \(q\) belonging to some separable Banach manifold \(X\). Each operator \(A(q)\) has a discrete spectrum of finite multiplicity without finite accumulation points. Under an appropriate transversality condition close to the strong Arnold hypothesis [V. I. Arnold, Funkts. Anal. Prilozh. 6, No. 2, 12-20 (1972; Zbl 0251.70012)] he proves that the members of the family \(A(q)\) having multiple eigenvalues form a manifold of codimension at least 2. Two examples are considered:
\(1^{0}\) Laplace operator with the domain as the parameter, \(2^{0}\) Schrödinger operator with the symmetric potential as the parameter.

MSC:

47A75 Eigenvalue problems for linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)

Citations:

Zbl 0251.70012
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References:

[1] ; Transversal mappings and flows. W. A. Benjamin, Inc., New York-Amsterdam, 1967.
[2] Arnold, Funkcional Anal i Prilo?en 6 pp 12– (1972)
[3] Besson, Comment Math Helv 64 pp 542– (1989)
[4] Colin de Verdiére, Comment Math Helv 63 pp 184– (1988)
[5] Garabedian, J Analyse Math 2 pp 281– (1953)
[6] ; Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Academic Press, New York-London, 1968.
[7] Lax, Comm Pure Appl Math 9 pp 747– (1956)
[8] Müller, Comm Pure Appl Math 7 pp 505– (1954)
[9] Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz. Gordon and Breach Science Publishers, New York-London-Paris, 1969.
[10] ; Functional analysis. Translated from the second French edition by Leo F. Boron. Reprint of the 1955 original. Dover Books on Advanced Mathematics. Dover Publications Inc New York, 1990.
[11] Degeneracies in the spectra of self-adjoint operators. Doctoral dissertation, Courant Institute of Mathematical Sciences, 1996.
[12] von Neumann, Z Phys A 30 pp 467– (1929)
[13] Uhlenbeck, Amer J Math 98 pp 1059– (1976)
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