Annales Henri Poincaré

, Volume 15, Issue 9, pp 1655–1696 | Cite as

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

Article

Abstract

The overlap, \({\mathcal{D}_N}\), between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\). We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\). The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\), and \({\tilde{\gamma}}\). In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.

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© Springer Basel 2013

Authors and Affiliations

  • Heinrich Küttler
    • 1
  • Peter Otte
    • 2
  • Wolfgang Spitzer
    • 3
  1. 1.Mathematisches Institut, LMU MünchenMunichGermany
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  3. 3.Fakultät für Mathematik und InformatikFernuniversität in HagenHagenGermany

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