THE SPECTRAL MINIMUM FOR RANDOM DISPLACEMENT MODELS

GUNTER STOLTZ

stolz@vorteb.math.uab.edu

Department of Mathematics

University of Alabama at Birmingham

University Station

Birmingham AL 35294, USA

University of Alabama at Birmingham

University Station

Birmingham AL 35294, USA

Consider a one-dimensional Schrödinger operator with potential
given as follows: Fix a *single site potential* which is
supported in an interval of length less than . Construct by
placing a translate of into each unit interval for
integer , where otherwise the positions of each translate are
arbitrary. Which configuration of single sites minimizes the spectral
minimum of the Schrödinger operator with potential ? This question
is equivalent to finding the spectral minimum of the random
displacement model. We conjecture that the minimum is realized through
*pair formation* of the single sites. We provide a partial proof of
this conjecture and additional numerical evidence for its correctness.
This is joint work with Jason Lott.

EPSRC Gregynog Workshop, 21-26 July 2002