Inverse spectral problem with partial information given on the potential and norming constants

Wei, Guangsheng; Xu, Hong-Kun

doi:10.1090/S0002-9947-2011-05545-5

Inverse spectral problem with partial information given on the potential and norming constants
HTML articles powered by AMS MathViewer

by Guangsheng Wei and Hong-Kun Xu PDF

Trans. Amer. Math. Soc. 364 (2012), 3265-3288 Request permission

Abstract:

The inverse spectral problem for a Sturm-Liouville equation in Liouville form with separated self-adjoint boundary conditions on the unit interval $[0,1]$ is considered. Some uniqueness results are obtained which imply that the potential $q$ can be completely determined even if only partial information is given on $q$ together with partial information on the spectral data, consisting of either one full spectrum and a subset of norming constants or a subset of pairs of eigenvalues and the corresponding norming constants. Moreover, the problem of missing eigenvalues and norming constants is also investigated in the situation where the potential $q$ is $C^{2k-1}$ and the boundary conditions at the endpoints $0$ and $1$ are fixed.

References

F. V. Atkinson, On the location of the Weyl circles, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 345–356. MR 616784, DOI 10.1017/S0308210500020163
L. Amour and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with potentials in $L^p$ spaces, Inverse Problems 23 (2007), no. 6, 2367–2373. MR 2441008, DOI 10.1088/0266-5611/23/6/006
L. Amour, J. Faupin, and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 (2009), no. 3, 033505, 14. MR 2510910, DOI 10.1063/1.3087426
Göran Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96 (German). MR 15185, DOI 10.1007/BF02421600
Göran Borg, Uniqueness theorems in the spectral theory of $y''+(\lambda -q(x))y=0$, Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, Johan Grundt Tanums Forlag, Oslo, 1952, pp. 276–287. MR 0058063
Carol Flynn Coleman and Joyce R. McLaughlin, Solution of the inverse spectral problem for an impedance with integrable derivative. I, II, Comm. Pure Appl. Math. 46 (1993), no. 2, 145–184, 185–212. MR 1199197, DOI 10.1002/cpa.3160460203
Carol Flynn Coleman and Joyce R. McLaughlin, Solution of the inverse spectral problem for an impedance with integrable derivative. I, II, Comm. Pure Appl. Math. 46 (1993), no. 2, 145–184, 185–212. MR 1199197, DOI 10.1002/cpa.3160460203
Björn E. J. Dahlberg and Eugene Trubowitz, The inverse Sturm-Liouville problem. III, Comm. Pure Appl. Math. 37 (1984), no. 2, 255–267. MR 733718, DOI 10.1002/cpa.3160370205
A. A. Danielyan and B. M. Levitan, Asymptotic behavior of the Weyl-Titchmarsh $m$-function, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 469–479 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 487–496. MR 1072691, DOI 10.1070/IM1991v036n03ABEH002031
Rafael del Rio, Fritz Gesztesy, and Barry Simon, Inverse spectral analysis with partial information on the potential. III. Updating boundary conditions, Internat. Math. Res. Notices 15 (1997), 751–758. MR 1470376, DOI 10.1155/S1073792897000494
W. N. Everitt, On a property of the $m$-coefficient of a second-order linear differential equation, J. London Math. Soc. (2) 4 (1971/72), 443–457. MR 298104, DOI 10.1112/jlms/s2-4.3.443
G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001. MR 2094651
Fritz Gesztesy and Barry Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267–297. MR 1616422, DOI 10.1007/BF02788147
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. I. The case of an a.c. component in the spectrum, Helv. Phys. Acta 70 (1997), no. 1-2, 66–71. Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Zürich, 1995). MR 1441597
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
Hald, O.H. Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc., 62, 1980, pp. 41-48.
Ole H. Hald, The inverse Sturm-Liouville problem with symmetric potentials, Acta Math. 141 (1978), no. 3-4, 263–291. MR 505878, DOI 10.1007/BF02545749
Harry Hochstadt, The inverse Sturm-Liouville problem, Comm. Pure Appl. Math. 26 (1973), 715–729. Collection of articles dedicated to Wilhelm Magnus. MR 330607, DOI 10.1002/cpa.3160260514
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
E. L. Isaacson and E. Trubowitz, The inverse Sturm-Liouville problem. I, Comm. Pure Appl. Math. 36 (1983), no. 6, 767–783. MR 720593, DOI 10.1002/cpa.3160360604
E. L. Isaacson, H. P. McKean, and E. Trubowitz, The inverse Sturm-Liouville problem. II, Comm. Pure Appl. Math. 37 (1984), no. 1, 1–11. MR 728263, DOI 10.1002/cpa.3160370102
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
Miklós Horváth, Inverse spectral problems and closed exponential systems, Ann. of Math. (2) 162 (2005), no. 2, 885–918. MR 2183284, DOI 10.4007/annals.2005.162.885
Levitan, B., On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl., 68, 1968, pp. 1-20.
B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 3–63 (Russian). MR 0162996
B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac operators, Mathematics and its Applications (Soviet Series), vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian. MR 1136037, DOI 10.1007/978-94-011-3748-5
V. A. Marčenko, Some questions of the theory of one-dimensional linear differential operators of the second order. I, Trudy Moskov. Mat. Obšč. 1 (1952), 327–420 (Russian). MR 0058064
Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
Joyce R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev. 28 (1986), no. 1, 53–72. MR 828436, DOI 10.1137/1028003
Jürgen Pöschel and Eugene Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, vol. 130, Academic Press, Inc., Boston, MA, 1987. MR 894477
Rafael del rio Castillo, On boundary conditions of an inverse Sturm-Liouville problem, SIAM J. Appl. Math. 50 (1990), no. 6, 1745–1751. MR 1080519, DOI 10.1137/0150103
Takashi Suzuki, Inverse problems for heat equations on compact intervals and on circles. I, J. Math. Soc. Japan 38 (1986), no. 1, 39–65. MR 816222, DOI 10.2969/jmsj/03810039
Guangsheng Wei and Hong-Kun Xu, On the missing eigenvalue problem for an inverse Sturm-Liouville problem, J. Math. Pures Appl. (9) 91 (2009), no. 5, 468–475 (English, with English and French summaries). MR 2517783, DOI 10.1016/j.matpur.2009.01.007
Anton Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. MR 2170950, DOI 10.1090/surv/121