The spectral minimum for random displacement models

摘要

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 an 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.