Asymptotic approximation for matrix differential equations and applications (Q1374504)

The authors develop a complex-valued and a real-valued vector and matrix extension of the scalar Liouville-Green asymptotic approximation. They consider the vector and matrix differential equations \(y''=A(t)y\) and \(Y''=A(t)Y\) on \([a,\infty ]\), where \(A(t)\) is a Hermitian, \(n\times n\) matrix function, analytic on \([a,\infty ]\) and invertible at \(\infty\). Analyticity of \(A(t)\) is a technical assumption but invertibility of \(A(t)\) is essential and has an analogy in the scalar case. They derive an asymptotic formula for \(Y\), \(Y'\) and an asymptotic expression for a determinant that determines the conjugate points of the above equations. They present the relation between the minimal negative eigenvalue of \(A(t)\) at \(t=\infty\) and the first conjugate point of \(a\) for large values of \(a\).