Abstract
This paper is the continuation of our work with Victor Guillemin (previous paper in this volume); Victor and I proved that the Taylor expansion of the potential at a generic non-degenerate critical point is determined by the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Here I show that, under some genericity assumptions, the potential of the 1D Schroedinger operator is determined by its semi-classical spectrum. Moreover, there is an explicit reconstruction. This paper is strongly related to a paper of David Gurarie (J. Math. Phys. 36:1934–1944 (1995)).
Mathematics Subject Classification (2010): 34E20, 81Q10, 81Q20
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Niels Abel. Auflösung einer mechanichen Aufgabe. Journal de Crelle 1 :153–157 (1826).
Marcel Berger, Paul Gauduchon and Edmond Mazet. Le spectre d’une variété riemannienne comapcte. Springer Lecture Notes in Mathematics, Vol. 194 (1971).
Yves Colin de Verdière. Semi-classical analysis and passive imaging. Nonlinearity 22 :45–75 (2009).
Yves Colin de Verdière. Bohr–Sommerfeld rules to all orders. Ann. Henri Poincaré 6 :925–936 (2005).
Yves Colin de Verdière.Spectrum of the Laplace operators and periodic geodesics:thirty years after. Ann. Inst. Fourier 57 :2429–2463 (2007).
Yves Colin de Verdière and Victor Guillemin. A semi-classical inverse problem I: Taylor expansions. Geometric Aspects of Analysis and Mechanics: In Honor of the 65th Birthday of Hans Duistermaat, (Eds) J. A. C. Kolk and Erik P. van den Ban, Birkh¨auser, Boston, MA, 81–95 (2011).
Eric Delabaere and Frédéric Pham. Unfolding the quartic oscillator. Annals of Physics 261 :180–218 (1997).
Alfonso Gracia-Saz. The symbol of a function of a pseudo-differential operator. Ann. Inst. Fourier 55 :2257–2284 (2005).
David Gurarie. Semi-classical eigenvalues and shape problems on surface of revolution. J. Math. Phys. 36 :1934–1944 (1995).
Bernard Helffer and Johannes Sjöstrand. Multiple wells in the semi-classical limit I, Commun. in PDE, 9 (4):337–408 (1984).
Mark Kac. Can one hear the shape of a drum? Amer. Math. Monthly, 73 (4):1–23 (1966).
B. M. Levitan and M. G. Gasymov. Determination of a differential equation by two of its spectra. Russian Math. Surveys 19 (2):1–63 (1964).
André Martinez. An Introduction to semi-classical Theory. Springer (2002).
Steve Zelditch. The inverse spectral problem. Surveys in Differential Geometry IX :401–467 (2004).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
To the memory of our friend, the inspiring mathematician, Hans Duistermaat
Rights and permissions
Copyright information
© 2011 Springer Science+Buisness Media, LLC
About this chapter
Cite this chapter
de Verdière, Y.C. (2011). A semi-classical inverse problem II: reconstruction of the potential. In: Kolk, J., van den Ban, E. (eds) Geometric Aspects of Analysis and Mechanics. Progress in Mathematics, vol 292. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8244-6_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8244-6_4
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-8243-9
Online ISBN: 978-0-8176-8244-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)