Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra

Eckhardt, Jonathan; Teschl, Gerald

doi:10.1090/S0002-9947-2012-05821-1

Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra
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by Jonathan Eckhardt and Gerald Teschl PDF

Trans. Amer. Math. Soc. 365 (2013), 3923-3942 Request permission

Abstract:

We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt–Lieberman type result for these operators. Our approach is based on the singular Weyl–Titchmarsh–Kodaira theory which is extended to cover the present situation.

References

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
D. S. Chelkak, Asymptotics of the spectral data of a harmonic oscillator perturbed by a potential with finite energy, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 31, 223–271, 325 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 129 (2005), no. 4, 4053–4082. MR 2037541, DOI 10.1007/s10958-005-0342-5
Dmitri Chelkak, Pavel Kargaev, and Evgeni Korotyaev, An inverse problem for an harmonic oscillator perturbed by potential: uniqueness, Lett. Math. Phys. 64 (2003), no. 1, 7–21. MR 1997715, DOI 10.1023/A:1024985302559
Dmitri Chelkak, Pavel Kargaev, and Evgeni Korotyaev, Inverse problem for harmonic oscillator perturbed by potential, characterization, Comm. Math. Phys. 249 (2004), no. 1, 133–196. MR 2077254, DOI 10.1007/s00220-004-1105-8
J. Eckhardt, Inverse uniqueness results for Schrödinger operators using de Branges theory, Complex Anal. Oper. Theory (to appear), DOI: 10.1007/s11785-012-0265-3.
Charles Fulton, Titchmarsh-Weyl $m$-functions for second-order Sturm-Liouville problems with two singular endpoints, Math. Nachr. 281 (2008), no. 10, 1418–1475. MR 2454944, DOI 10.1002/mana.200410689
C. Fulton and H. Langer, Sturm-Liouville operators with singularities and generalized Nevanlinna functions, Complex Anal. Oper. Theory 4 (2010), no. 2, 179–243. MR 2652523, DOI 10.1007/s11785-009-0026-0
F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), no. 2, 401–446. MR 1244942, DOI 10.1006/jfan.1993.1132
F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc. 348 (1996), no. 1, 349–373. MR 1329533, DOI 10.1090/S0002-9947-96-01525-5
Fritz Gesztesy and Barry Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2765–2787. MR 1694291, DOI 10.1090/S0002-9947-99-02544-1
F. Gesztesy, B. Simon, and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators, J. Anal. Math. 70 (1996), 267–324. MR 1444263, DOI 10.1007/BF02820446
Fritz Gesztesy and Maxim Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr. 279 (2006), no. 9-10, 1041–1082. MR 2242965, DOI 10.1002/mana.200510410
Harry Hochstadt and Burton Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), no. 4, 676–680. MR 470319, DOI 10.1137/0134054
Lars Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943–949. MR 226387, DOI 10.1090/S0002-9904-1967-11860-3
Miklós Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 (2001), no. 10, 4155–4171. MR 1837225, DOI 10.1090/S0002-9947-01-02765-9
Rostyslav O. Hryniv and Yaroslav V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems 20 (2004), no. 5, 1423–1444. MR 2109127, DOI 10.1088/0266-5611/20/5/006
I. S. Kac, The existence of spectral functions of generalized second order differential systems with boundary conditions at the singular end, AMS Translations (2) 62, 204–262 (1967).
A. Khodakovsky, Inverse spectral problem with partial information on the potential: the case of the whole real line, Comm. Math. Phys. 210 (2000), no. 2, 399–411. MR 1776838, DOI 10.1007/s002200050785
Kunihiko Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of $S$-matrices, Amer. J. Math. 71 (1949), 921–945. MR 33421, DOI 10.2307/2372377
Aleksey Kostenko and Gerald Teschl, On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators, J. Differential Equations 250 (2011), no. 9, 3701–3739. MR 2773183, DOI 10.1016/j.jde.2010.10.026
Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl, Inverse eigenvalue problems for perturbed spherical Schrödinger operators, Inverse Problems 26 (2010), no. 10, 105013, 14. MR 2719774, DOI 10.1088/0266-5611/26/10/105013
Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. IMRN 8 (2012), 1699–1747. MR 2920828, DOI 10.1093/imrn/rnr065
A. Kostenko, A. Sakhnovich, and G. Teschl, Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr. 285, 392–410 (2012).
Pavel Kurasov and Annemarie Luger, An operator theoretic interpretation of the generalized Titchmarsh-Weyl coefficient for a singular Sturm-Liouville problem, Math. Phys. Anal. Geom. 14 (2011), no. 2, 115–151. MR 2802457, DOI 10.1007/s11040-011-9090-6
B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006, DOI 10.1090/mmono/150
B. M. Levitan, Sturm-Liouville operators on the entire real axis with the same discrete spectrum, Mat. Sb. (N.S.) 132(174) (1987), no. 1, 73–103, 143 (Russian); English transl., Math. USSR-Sb. 60 (1988), no. 1, 77–106. MR 883914, DOI 10.1070/SM1988v060n01ABEH003157
H. P. McKean and E. Trubowitz, The spectral class of the quantum-mechanical harmonic oscillator, Comm. Math. Phys. 82 (1981/82), no. 4, 471–495. MR 641910
L. Sakhnovich, Half-inverse problems on the finite interval, Inverse Problems 17 (2001), no. 3, 527–532. MR 1843279, DOI 10.1088/0266-5611/17/3/311
Gerald Teschl, Mathematical methods in quantum mechanics, Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009. With applications to Schrödinger operators. MR 2499016, DOI 10.1090/gsm/099
Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320, DOI 10.1007/BFb0077960
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759