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.


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


Zbl 0251.70012
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