Annales Henri Poincaré

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

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

  • Heinrich Küttler
  • Peter OtteEmail author
  • Wolfgang Spitzer


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.


Fermi Energy Thermodynamic Limit Integral Formula Spectral Projection Spectral Shift Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Authors and Affiliations

  • Heinrich Küttler
    • 1
  • Peter Otte
    • 2
    Email author
  • 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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