The spectral minimum for random displacement models

Consider a one-dimensional Schrr 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 connguration of single sites minimizes the spectral minimum of the Schrr odinger operator with potential V ? This question is equivalent to nding 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.