Extremal Problems for Eigenvalue Functionals

We consider the eigenvalues, $\lambda _n (\rho )$, of self-adjoint Sturm–Liouville systems to be real valued functionals of certain coefficient functions in the differential equation. We introduce a classical (in general nonlinear) functional $K (\rho )$ which is tangent to $\lambda _n (\rho )$ at a fixed function $\rho ^ * $. That is, $\lambda _n (\rho ^ * ) = K(\rho ^ * )$ and $\delta \lambda _n = \delta K$ at $\rho ^ * $. Then by using classical calculus of variations on $K (\rho )$ we show how to find extremals of $\lambda _n (\rho )$ over certain classes of functions p.