Abstract
We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L 1([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.
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Andrew A.L.: Computing Sturm-Liouville potentials from two spectra. Inverse Problems 22(6), 2069–2081 (2006)
Brown B.M., Knowles I., Weikard R.: On the inverse resonance problem. J. London Math. Soc. (2) 68(2), 383–401 (2003)
Brown B.M., Samko V.S., Knowles I.W., Marletta M.: Inverse spectral problem for the Sturm-Liouville equation. Inverse Problems 19(1), 235–252 (2003)
Brown B.M., Weikard R.: The inverse resonance problem for perturbations of algebro-geometric potentials. Inverse Problems 20(2), 481–494 (2004)
Chadan K., Sabatier P.C.: Inverse Problems in Quantum Scattering Theory. Second edition. Texts and Monographs in Physics. Springer-Verlag, New York (1989) (With a foreword by R. G. Newton)
Hald O.H.: The inverse Sturm-Liouville problem with symmetric potentials. Acta Math. 141(3-4), 263–291 (1978)
Hitrik M.: Stability of an inverse problem in potential scattering on the real line. Comm. Part. Diff. Eqs. 25(5–6), 925–955 (2000)
Korotyaev E.: Inverse resonance scattering on the half line. Asymptot. Anal. 37(3–4), 215–226 (2004)
Korotyaev E.: Stability for inverse resonance problem. Int. Math. Res. Not. 73, 3927–3936 (2004)
Levitan, B.M.: Inverse Sturm-Liouville problems. Zeist: VSP, 1987 (translated from the Russian by O. Efimov)
Marchenko, V.A.: Sturm-Liouville Operators and Applications. Volume 22 of Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag 1986 (translated from the Russian by A. Iacob.)
Marletta M., Weikard R.: Weak stability for an inverse Sturm-Liouville problem with finite spectral data and complex potential. Inverse Problems 21(4), 1275–1290 (2005)
Marletta M., Weikard R.:: Stability for the inverse resonance problem for a Jacobi operator with complex potential. Inverse Problems 23(4), 1677–1688 (2007)
Naĭmark, M.A.: Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, With additional material by the author, and a supplement by V.È. Ljance, Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt. New York: Frederick Ungar Publishing Co., 1968
Newton, R.G.: Scattering Theory of Waves and Particles. Mineola, NY: Dover Publications Inc., 2002, reprint of the 1982 second edition New York: Springer, with list of errata prepared for this edition by the author
Paine J.: A numerical method for the inverse Sturm-Liouville problem. SIAM J. Sci. Statist. Comput. 5(1), 149–156 (1984)
Röhrl N.: A least-squares functional for solving inverse Sturm-Liouville problems. Inverse Problems 21(6), 2009–2017 (2005)
Rundell W., Sacks P.E.: Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comp. 58(197), 161–183 (1992)
Weißkopf V., Wigner E.: Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. f. Physik 63, 54–73 (1930)
Zworski M.: Resonances in physics and geometry. Notices Amer. Math. Soc. 46(3), 319–328 (1999)
Zworski M.: A remark on isopolar potentials. SIAM J. Math. Anal. 32(6), 1324–1326 (electronic), (2001)
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Marlettta, M., Shterenberg, R. & Weikard, R. On the Inverse Resonance Problem for Schrödinger Operators. Commun. Math. Phys. 295, 465–484 (2010). https://doi.org/10.1007/s00220-009-0928-8
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DOI: https://doi.org/10.1007/s00220-009-0928-8