Henrik Ueberschaer

Publications (6) View all

• Article: Quantum Chaos for point scatterers on flat tori

  Henrik Ueberschaer
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  ABSTRACT: This survey article deals with a delta potential - also known as a point scatterer - on flat 2D and 3D tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so- called weak and strong coupling regimes. We report on recent progress as well as a number of open problems in this field.
  12/2012;
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  Available from: Henrik Ueberschaer

  Article: On the eigenvalue spacing distribution for a point scatterer on the flat torus

  Zeev Rudnick, Henrik Ueberschaer
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  ABSTRACT: We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of the Laplacian (ignoring multiplicties), by showing that the perturbed eigenvalues generically clump with the unperturbed ones on the scale of the mean level spacing. We also study the three dimensional case, where the situation is very different.
  08/2012;
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  Available from: ArXiv

  Article: The trace formula for a point scatterer on a hyperbolic surface with one cusp

  Henrik Ueberschaer
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  ABSTRACT: We prove an exact trace formula for the Laplacian with a delta potential - also known as a point scatterer - on a non-compact hyperbolic surface of finite volume with one cusp. Our formula is an analogue of the Selberg trace formula. We express the difference of perturbed and unperturbed trace as a smooth term plus a sum over combinations of diffractive orbits.
  11/2011;
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  Available from: ArXiv

  Article: Statistics of wave functions for a point scatterer on the torus

  Zeev Rudnick, Henrik Ueberschaer
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  ABSTRACT: Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.
  09/2011;
• Source

  Available from: ArXiv

  Article: The trace formula for a point scatterer on a compact hyperbolic surface

  Henrik Ueberschaer
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  ABSTRACT: An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.
  09/2011;