Rendiconti del Circolo Matematico di Palermo Series 2

, Volume 38, Issue 3, pp 329–370

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

• G. Baur
• W. Kratz
Article

Abstract

For given 2n×2n matricesS13,S24 with rank(S13,S24)=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′=C1(x;λ)u-AT(x)v with
$$S_{13} \left( {_{u\left( b \right)}^{ - u\left( a \right)} } \right) + S_{24} \left( {_{\upsilon \left( b \right)}^{\upsilon \left( a \right)} } \right) = 0,{\text{ }}a< b;$$
where we assume that then×n matrices,A, B, C1 satisfy:A, B, C1, ∂/∂λC1 are continuous on IR resp. IR2;B, C1 are Hermitian;B, −∂/∂λC1 are non-negative definite; and we assume the crucial normality-condition: for any solutionu, v (λ∈IR arbitrary) ∂/∂λC1u≡0 on some interval always impliesuv≡0. Then, the main result of the paper (Theorem 2) is the following oscillation result: For any conjoined basisU1(x; λ),V1(x; λ) of the differential system with fixed (with respect to λ) initial valuesU1(a), V1(a), we haven1(λ)+n2(λ)=n3(λ)+n1+n2 for λ ∈ IR with regularU1(b; λ); where$$n_i = \mathop {lim}\limits_{\lambda \to \infty }$$;ni(λ),i=1,2;n1(λ) denotes number of focal points of 3U1 in [a, b);n3(λ) denotes the number of eigenvalues which are ≤λ; andn2(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).

Keywords

Eigenvalue Problem Hamiltonian System Differential System Riccati Equation Matrix Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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