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Inverse Spectral Problems for Schrödinger Operators

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Abstract

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).

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  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Hamid Hezari

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  1. Hamid Hezari
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Correspondence to Hamid Hezari.

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Communicated by B. Simon

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Hezari, H. Inverse Spectral Problems for Schrödinger Operators. Commun. Math. Phys. 288, 1061–1088 (2009). https://doi.org/10.1007/s00220-008-0718-8

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  • DOI: https://doi.org/10.1007/s00220-008-0718-8

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