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

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 impliesu≡v≡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).