Linear independence of the principal solutions at ∞ and −∞ for formally self-adjoint differential systems

AbstractLinear independence of the principal solutions at ∞ and −∞ for identically normal, formally self-adjoint differential systems U′ = AU + BV, V′ = CU − A∗V, with B ⩾ 0, which are disconjugate on (−∞, ∞), is characterized by the condition W−∞ < W∞ on (−∞, ∞), where W∞, [W−∞], is the distinguished solution of the associated Riccati equation at ∞, [−∞]. A comparison type theorem between the above system and certain Sturmian majorants gives order relations for the corresponding distinguished solutions. The work is motivated by that of Anderson and Lazer for self-adjoint scalar equations of order 2n. For certain types of such equations, the results allow a direct sum decomposition into two n dimensional families, one whose solutions are in L2(−∞, 0] and the other whose solutions are in L2[0, ∞)