The decomposition of \(M(\lambda)\) surfaces using Niessen's limit circles
There are two approaches in extending Weyl's theorem concerning the existence of square integrable solutions of a homogeneous differential equation of second order \(-(py')'+qy=\lambda wy\) to higher order or Hamiltonian systems of even dimension greater than or equal to 2, say, systems of the form \(JY'=(\lambda A+B)Y\), where Y is of dimension 2n, \(A=A^*\geq 0\), \(B=B^*\) are real, locally integrable 2n\(\times 2n\) matrices, and \(J=\left[ \begin{matrix} 0\\ I_ n\end{matrix} \begin{matrix} -I_ n\\ 0\end{matrix} \right]\). One approach by Hinton-Shaw uses a matrix M(\(\lambda\)) to generate square integrable solutions; another approach by Niessen is to use the eigenvalues of a symmetric monotonic matrix to generate square integrable solutions. The paper connects the \(n\times n\) matrix M(\(\lambda\)) of Hinton-Shaw with the n two dimensional circles of Niessen. It is shown that M(\(\lambda\)) is represented by ellipsoids which involve the Niessen eigenvalues and circle radii.
homogeneous differential equation of second order
scientific article; zbMATH DE number 4202801
scientific article