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A limit law for the ground state of Hill's equation

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Abstract

It is proved that the ground state Λ(L) of (−1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law

$$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$

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Authors and Affiliations

  1. Courant Institute of Mathematical Science, New York, New York

    H. P. McKean

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  1. H. P. McKean
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McKean, H.P. A limit law for the ground state of Hill's equation. J Stat Phys 74, 1227–1232 (1994). https://doi.org/10.1007/BF02188225

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  • DOI: https://doi.org/10.1007/BF02188225

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