Periodic potentials with minimal energy bands

Abstract:

We consider the problem of minimizing the width of the lowest band in the spectrum of Hill’s equation, $- u'' + q\left ( x \right )u = \lambda u$ on $\mathbb {R}$ with $q\left ( {x + 1} \right ) = q\left ( x \right )$ for all $x \in \mathbb {R}$, when the potential function $q$ is allowed to vary over a ball of radius $M > 0{\text { in }}{L^\infty }$. We show that minimizing potentials ${q_ * }$ exist and that, when considered as functions on the circle, they must have exactly one well on which ${q_ * }\left ( x \right )$ must equal $- M$ and one barrier on which ${q_ * }\left ( x \right )$ must equal $M$; these are the only values that ${q_ * }$ can assume (up to changes on sets of measure zero). That is, on the circle there is a single interval where ${q_ * }\left ( x \right ) = M$ and on the complementary interval ${q_ * }\left ( x \right ) = - M$. These results can be used to solve the problem of minimizing the gap between the lowest Neumann eigenvalue and either the lowest Dirichlet eigenvalue or the second Neumann eigenvalue for the same equation restricted to the interval $[0,1]$.