Abstract
For given 2n×2n matricesS 13,S 24 with rank(S 13,S 24)=2n \(S_{13} \bar S_{24}^T = S_{24} \bar S_{13}^T \) we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C 1(x;λ)u-A T(x)v with
where we assume that then×n matrices,A, B, C 1 satisfy:A, B, C 1, ∂/∂λC 1 are continuous on IR resp. IR2;B, C 1 are Hermitian;B, −∂/∂λC 1 are non-negative definite; and we assume the crucial normality-condition: for any solutionu, v (λ∈IR arbitrary) ∂/∂λC 1 u≡0 on some interval always impliesu≡v≡0. Then, the main result of the paper (Theorem 2) is the following oscillation result: For any conjoined basisU 1(x; λ),V 1(x; λ) of the differential system with fixed (with respect to λ) initial valuesU 1(a), V1(a), we haven 1(λ)+n 2(λ)=n 3(λ)+n 1+n 2 for λ ∈ IR with regularU 1(b; λ); where\(n_i = \mathop {lim}\limits_{\lambda \to \infty } \);n i(λ),i=1,2;n 1(λ) denotes number of focal points of 3U 1 in [a, b);n 3(λ) denotes the number of eigenvalues which are ≤λ; andn 2(λ) denotes the number of negative eigenvalues of a certain Hermitian 3n×3n matrixM(λ). Moreover, it is shown how classical results (e.g. Rayleigh's principle, existence theorem) can be derived from this oscillation theorem via a generalized Picone identity (which yields also the matrixM(λ) above).
Actually these eigenvalue problems in connection with an associated functional (the linear differential system above consists of the canonical form of the Euler-Lagrange equations of a corresponding Bolza problem) are very much related to the work of W.T. Reid (Wiley 1971). Many results of this paper, including the oscillation theorem above, are extensions of an earlier paper on Sturm-Liouville eigenvalue problems (Analysis 5(1985), 97–152).
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Baur, G., Kratz, W. A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals. Rend. Circ. Mat. Palermo 38, 329–370 (1989). https://doi.org/10.1007/BF02850019
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DOI: https://doi.org/10.1007/BF02850019