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A semi-classical inverse problem. I: Taylor expansions. (English) Zbl 1258.34036

van den Ban, Erik P. (ed.) et al., Geometric aspects of analysis and mechanics. In honor of the 65th birthday of Hans Duistermaat. Basel: Birkhäuser (ISBN 978-0-8176-8243-9/hbk; 978-0-8176-8244-6/ebook). Progress in Mathematics 292, 81-95 (2011).
The authors 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. In contrast to previous works, in this paper the authors do not assume that the potential is even.
For the entire collection see [Zbl 1218.00014].

MSC:

34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
47G30 Pseudodifferential operators
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