02-12 Jason Lott, G\"unter Stolz

The spectral minimum for random displacement models (564K, Postscript) Jan 8, 02

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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.

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