Abstract
We consider an operator function F defined on the interval \(\user2{[}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}\) whose values are semibounded self-adjoint operators in the Hilbert space \(\mathfrak{H}\). To the operator function F we assign quantities \(\mathcal{N}_\user1{F}\) and ν F (λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval [σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ [σ,τ]. We present conditions under which the estimate \(\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}\) holds. We also establish conditions for the relation \(\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}\) to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.
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Vladimirov, A.A. Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions. Mathematical Notes 74, 794–802 (2003). https://doi.org/10.1023/B:MATN.0000009015.40046.63
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DOI: https://doi.org/10.1023/B:MATN.0000009015.40046.63