A semi-classical inverse problem I: Taylor expansions

Part of the Progress in Mathematics book series (PM, volume 292)


In dimension 1, we show that the Taylor expansion of a “generic” potential near a nondegenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Contrary to the work of previous authors, we do not assume that the potential is even. The classical Birkhoff normal form does not contain enough information to determine the potential, but the quantum Birkhoff normal form does. In a companion paper [5], the first author shows how the potential itself is, without any analyticity assumption and under some mild genericity hypotheses, determined by the semi-classical spectrum.


Schrödinger operator semi-classics inverse spectral problem 


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© Springer Science+Buisness Media, LLC 2011

Authors and Affiliations

  1. 1.Institut FourierUnité mixte de recherche CNRS-UJF 5582Saint Martin d’Hères CedexFrance
  2. 2.Math. Dept.MITCambridgeUSA

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