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How rare are multiple eigenvalues?
Author(s) -
Teytel Mikhail
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199908)52:8<917::aid-cpa1>3.0.co;2-s
Subject(s) - mathematics , codimension , eigenvalues and eigenvectors , spectrum (functional analysis) , separable space , differentiable function , pure mathematics , hilbert space , operator (biology) , mathematical analysis , transversality , simple (philosophy) , banach space , laplace operator , multiplicity (mathematics) , physics , quantum mechanics , biochemistry , chemistry , philosophy , epistemology , repressor , transcription factor , gene
Let A ( q ) be a differentiable family of self‐adjoint operators on a Hilbert space H , indexed by a parameter q that belongs to a separable Banach manifold X . Assume that the spectrum of each operator A ( q ) is discrete, of finite multiplicity, and with no finite accumulation points. We introduce a new concept of codimension in infinite‐dimensional space and then prove that under an appropriate transversality condition, related to the strong Arnold hypothesis, the members of the family A ( q ) having multiple eigenvalues form a set of codimension at least 2. Using this, we show that a generic member of the family A ( q ) has a simple spectrum (i.e., no repeated eigenvalues) and that any two values q 1 and q 2 of the parameter can be connected by an analytic curve γ in X such that A ( q ) has a simple spectrum for all q in the interior of γ. We then apply these results in two cases of physical interest: to the Laplace operator with the domain as parameter and to the Schrödinger operator with a symmetric potential as parameter. © 1999 John Wiley & Sons, Inc.