Jason Lott, G\"unter Stolz
The spectral minimum for random displacement models
(564K, Postscript)
ABSTRACT. Consider a one-dimensional Schr\"odinger 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 Schr\"odinger 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 {\em pair formation} of the single sites. We
provide a partial proof of this conjecture and additional
numerical evidence for its correctness.