An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions

Pages: 933 - 978, Volume 24, Issue 3, July 2009

doi:10.3934/dcds.2009.24.933       Abstract        Full Text (388.1K)

Xing-Bin Pan - Department of Mathematics, East China Normal University, Shanghai 200062, China (email)

Abstract: 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.

Keywords:  liquid crystal, magnetic Schrödinger operator, lowest eigenvalue, eigenvalue variation, Landau-de Gennes model, critical wave number.
Mathematics Subject Classification:  Primary: 82D30, 82D55; Secondary: 35J10, 35P15, 35Q55.

Received: August 2007;      Revised: June 2008;      Published: April 2009.