Maximal and minimal eigenvalues and their associated nonlinear equations
Mark S. Ashbaugh, Evans M. Harrell · Journal of Mathematical Physics · 1987
The spectral theory of uniformly elliptic operators A under perturbations V giving rise to operators of the form HV=A+V(x) on a bounded or unbounded region, such as Schrödinger operators, are considered. Suppose that ∥V∥p is constrained, but V is otherwise unspecified. The theory of the potentials V that maximize or minimize the eigenvalues of HV is presented. The optimizing potentials are typically determined by equations of the form −Δu+W(x)u=±cuα+Λu. The optimization of eigenvalues also turns out to be related to the determination of the best constants in Sobolev’s inequality, and, in its one-dimensional simplification, to a classical oscillator problem with ‘‘instanton’’ properties.