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Coherent States and Applications in Mathematical Physics pp 123–150Cite as

Trace Formulas and Coherent States

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Abstract

The most known trace formula in mathematical physics is certainly the Gutzwiller trace formula linking the eigenvalues of the Schrödinger operator \(\hat{H}\) as Planck’s constant goes to zero (the semi-classical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of \(\hat{H}\). In mathematics this kind of trace formula was first known as Poisson formula. It was proved first for the Laplace operator on a compact manifold, then for more general elliptic operators using the theory of Fourier-integral operators. Our goal here is to show how the use of coherent states allows us to give a rather simple and direct rigorous proof.

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Notes

  1. 1.

    See the definition in Sect. 5.2.

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Authors and Affiliations

  1. Batiment Paul Dirac, IPNL, Villeurbanne, France

    Monique Combescure

  2. Laboratoire Jean-Leray Departement de Mathematiques, Nantes University, Nantes Cedex 03, France

    Didier Robert

Authors
  1. Monique Combescure
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  2. Didier Robert
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Combescure, M., Robert, D. (2012). Trace Formulas and Coherent States. In: Coherent States and Applications in Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0196-0_5

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