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from block copolymer morphology
An eigenvalue variation problem of magnetic
Schrödinger operator in three dimensions
This paper concerns the lowest eigenvalue $\mu(b\N^Q)$ of the
Schrödinger operator in three-dimensions with a magnetic potential
$b\N^Q$, where the vector field $\N^Q$ depends on a matrix $Q$
varying in $SO(3)$ and $b$ is a real parameter. The eigenvalue
variation problem is to minimize the lowest eigenvalue among all $Q$
in $SO(3)$. This problem arises in the phase transitions of smectic
liquid crystals. We give an estimate of the minimum value
inf${\mu(b\N^Q):~Q\in SO(3)\}$ for large $b$, and examine its
dependence on geometry of the domain surface.