On a generalized sturm theorem

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm type oscillation theorem for indefinite systems with Dirichlet boundary conditions of the form p2m d2mu/dx2m + p2m-2(x) d2m-2u/dx2m-2 + ⋯ + p1 (x) du/dx + po (x) u = 0, where p_i is a smooth path of matrices on the complex n-dimensional vector space ℂn, p_2m is the symmetry represented by the diagonal block matrix diag (I_n-v, -I_v), and where v is an integer between 0 and n and I is the identity matrix.