An eigenvalue variation problem of magneticSchrödinger operator in three dimensions

Xing-Bin Pan

doi:10.3934/dcds.2009.24.933

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2009, Volume 24, Issue 3: 933-978. Doi: 10.3934/dcds.2009.24.933

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An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions

Xing-Bin Pan^1,

1.
Department of Mathematics, East China Normal University, Shanghai 200062

Received: July 31, 2007

Revised: May 31, 2008

Published: August 2009

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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.

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Mathematics Subject Classification: Primary: 82D30, 82D55; Secondary: 35J10, 35P15, 35Q55.

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