Geometric properties of the self-adjoint Sturm-Liouville problems (Q1771382)

Considering the (real) quadratic form \[ U=\frac12\int_0^L \left(p(x)(w'(x))^2+q(x)w(x)\right)dx, \] also called energy functional, associated with the selfadjoint Sturm-Liouville problem \[ (pw')'-qw+\lambda \rho w=0,\;0\leq x\leq L,\quad w(0)=w(L)=0, \] the author uses the variational principle to relate curvature lines and curvature of \(z=U\) to eigenvectors and eigenvalues, respectively, of the Sturm-Liouville problem. Hilbert spaces \(H_1\) and \(H_{11}\) occur, but are not defined explicitly. Most likely, they are certain Sobolev spaces with suitable boundary conditions.