Transactions of the American Mathematical Society

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

Author(s): Fritz Gesztesy; Barry Simon
Journal: Trans. Amer. Math. Soc. 352 (2000), 2765 - 2787.
MSC (2000): Primary 34A55, 34L40; Secondary 34B20
Posted: December 10, 1999
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Abstract: We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential

of a one-dimensional Schrödinger operator $H=-\frac{d^{2}}{dx^{2}}+q$ determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of

on a finite interval and knowledge of

over a corresponding fraction of the interval. The methods employed rest on Weyl

-function techniques and densities of zeros of a class of entire functions.

1.: F.V. Atkinson, On the location of the Weyl Circles, Proc. Roy. Soc. Edinburgh 88A 1981 345-356 MR 83a:34023
2.: G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math. 78 1946 1-96 MR 7:382d
3.: G. Borg, Uniqueness theorems in the spectral theory of $y''+(\lambda -q(x))y=0$ , Proc. 11th Scandinavian Congr. Math. (Trondheim, 1949), Johan Grundt Tanums Forlag, Oslo, 1952, 276-287. MR 15:315a
4.: E.A Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985. MR 16:1022b (earlier ed.)
5.: A.A. Danielyan and B.M. Levitan, On the asymptotic behavior of the Weyl-Titchmarsh -function, Math. USSR Izv. 36 1991 487-496 MR 92d:34056
6.: R. del Rio, F. Gesztesy, and B. Simon, Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Internat. Math. Res. Notices 1997, no.15, 751-758 MR 99a:34032
7.: R. del Rio, F. Gesztesy, and B. Simon, Corrections and addendum to inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Internat. Math. Res. Notices 1999, no.11, 623-625. CMP 99:14
8.: W.N. Everitt, On a property of the -coefficient of a second-order linear differential equation, J. London Math. Soc. 4 1972 443-457 MR 45:7156
9.: F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Higher order trace relations for Schrödinger operators Revs. Math. Phys. 7 1995 893-922 MR 97d:34094
10.: F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc. 348 1996 349-373 MR 96e:34030
11.: F. Gesztesy and B. 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 66-71 MR 98m:34171
12.: F. Gesztesy and B. Simon, -functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. d'Analyse Math. 73 (1997), 267-297. MR 99c:47039
13.: O.H. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc. 62 1980 41-48
14.: H. Hochstadt, The inverse Sturm-Liouville problem Commun. Pure Appl. Math. 26 1973 715-729 MR 48:8944
15.: H. Hochstadt, On the construction of a Jacobi matrix from mixed given data, Lin. Algebra Appl. 28 1979 113-115 MR 81a:15024
16.: H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 1978 676-680 MR 57:10077
17.: K. Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. Ser. 4 149 1987 185-206 MR 89d:34053
18.: S. Jayawardena, The reconstruction of symmetric, anti-symmetric and partially known potentials from spectral data, preprint.
19.: A.Ben Khaled, Problème inverse de Sturm-Liouville associé à un opérateur différentiel singulier, C. R. Acad. Sc. Paris Sér. I Math. 299 1984 221-224 MR 86f:34045
20.: M.G. Krein, On the solution of the inverse Sturm-Liouville problem, Dokl. Akad. Nauk SSSR 76 (1951), 21-24 (Russian). MR 12:613e
21.: B. Ja. Levin, Distribution of Zeros of Entire Functions (rev. ed.), Amer. Math. Soc., Providence, RI, 1980. MR 81k:30011
22.: N. Levinson, The inverse Sturm-Liouville problem Mat. Tidskr. B 1949 25-30 MR 11:248e
23.: B. Levitan, On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl. 68 1968 1-20 MR 28:3194
24.: B. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987. MR 89b:34001
25.: B.M. Levitan and M.G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surv. 19:2 (1964), 1-63. MR 29:299
26.: B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluwer, Dordrecht, 1991. MR 92i:34119
27.: M.M. Malamud, Similarity of Volterra operators and related questions of the theory of differential equations of fractional order, Trans. Moscow Math. Soc. 55 1994 57-122 MR 98e:47051
28.: V.A. Marchenko, Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy Moskov. Mat. Obsc. 1 (1952), 327-420 (Russian); English transl. in Amer. Math. Soc. Transl. (2) 101 (1973), 1-104. MR 15:315b
29.: V. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. MR 88f:34034
30.: A.I. Markushevich, Theory of Functions of a Complex Variable (2nd ed.), Chelsea, New York, 1985. MR 56:3258 (earlier ed.)
31.: J. Pöschel and E. Trubowitz, Inverse Scattering Theory, Academic Press, Boston, 1987. MR 89b:34061
32.: M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Analysis of Operators, Academic Press, New York, 1978. MR 58:12429c
33.: R. del Rio Castillo, On boundary conditions of an inverse Sturm-Liouville problem, SIAM J. Appl. Math. 50 1990 1745-1751 MR 91j:34023
34.: W. Rundell and P.E. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp. 58 1992 161-183 MR 92h:34034
35.: T. Suzuki, Deformation formulas and their applications to spectral and evolutional inverse problems, Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., vol. 81, North-Holland, Amsterdam, 1983, pp. 289-311. MR 85j:35196
36.: T. Suzuki, Gel'fand-Levitan's theory, deformation formulas and inverse problemsJ. Fac. Sci. Univ. Tokyo Sect. IA, Math. 32 1985 223-271 MR 86k:35157
37.: T. Suzuki, Inverse problems for heat equations on compact intervals and on circles, I, J. Math. Soc. Japan 38 1986 39-65 MR 87f:35241
38.: E.C. Titchmarsh, Eigenfunction Expansions, Part I (2nd ed.), Clarendon Press, Oxford, 1962. MR 31:426

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 34A55, 34L40, 34B20

Fritz Gesztesy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: fritz@math.missouri.edu

Barry Simon
Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
Email: bsimon@caltech.edu

DOI: 10.1090/S0002-9947-99-02544-1
PII: S 0002-9947(99)02544-1
Received by editor(s): October 9, 1997
Posted: December 10, 1999
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.
Copyright of article: Copyright 2000, by the Authors

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