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

Consider a one-dimensional Schrödinger operator with potential $V$ given as follows: Fix a single site potential $f$ which is supported in an interval of length less than $1$. Construct $V$ by placing a translate of $f$ into each unit interval $[n,n+1]$ for integer $n$, where otherwise the positions of each translate are arbitrary. Which configuration of single sites minimizes the spectral minimum of the Schrödinger operator with potential $V$? 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.