Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators
HTML articles powered by AMS MathViewer
- by F. Gesztesy and B. Simon
- Trans. Amer. Math. Soc. 348 (1996), 349-373
- DOI: https://doi.org/10.1090/S0002-9947-96-01525-5
Abstract:
New unique characterization results for the potential $V(x)$ in connection with Schrödinger operators on $\Bbb R$ and on the half-line $[0,\infty )$ are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.References
- Sergio Albeverio, Friedrich Gesztesy, Raphael Høegh-Krohn, and Helge Holden, Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer-Verlag, New York, 1988. MR 926273, DOI 10.1007/978-3-642-88201-2 N. Aronszajn and W.F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Anal. Math. 5 (1957), 321–388.
- 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
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985.
- Walter Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys. 126 (1989), no. 2, 379–407. MR 1027503, DOI 10.1007/BF02125131
- P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
- B. A. Dubrovin, A periodic problem for the Korteweg-de Vries equation in a class of short-range potentials, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 41–51 (Russian). MR 0486780
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- 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
- H. Flaschka, On the inverse problem for Hill’s operator, Arch. Rational Mech. Anal. 59 (1975), no. 4, 293–309. MR 387711, DOI 10.1007/BF00250422
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3 F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators, Acta Applicandae Math. 39 (1995), 315–333. F. Gesztesy and B. Simon, The xi function, Acta Math. (to appear). F. Gesztesy and B. Simon, Rank one perturbations at infinite coupling, J. Funct. Anal. 128 (1995), 245–252 (to appear). F. Gesztesy, H. Holden, and B. Simon, Absolute summability of the trace relation for certain Schrödinger operators, Commun. Math. Phys. 168 (1995), 137–161. F. Gesztesy, B. Simon, and G. Teschl, work in preparation.
- F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Trace formulae and inverse spectral theory for Schrödinger operators, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 250–255. MR 1215308, DOI 10.1090/S0273-0979-1993-00431-2
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94 H. Grosse and A. Martin, Theory of the inverse problem for confining potentials (I). Zero angular momentum, Nucl. Phys. B148 (1979), 413–432.
- Harry Hochstadt, On the determination of a Hill’s equation from its spectrum, Arch. Rational Mech. Anal. 19 (1965), 353–362. MR 181792, DOI 10.1007/BF00253484
- V. A. Javrjan, Regularized trace of the difference of two singular Sturm-Liouville operators, Dokl. Akad. Nauk SSSR 169 (1966), 49–51 (Russian). MR 0202007
- V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2–3, 246–251 (Russian, with Armenian and English summaries). MR 0301565 A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), 345–356.
- S. Kotani and M. Krishna, Almost periodicity of some random potentials, J. Funct. Anal. 78 (1988), no. 2, 390–405. MR 943504, DOI 10.1016/0022-1236(88)90125-5 M.G. Krein, Perturbation determinants and a formula for the traces of unitary and self-adjoint operators, Soviet Math. Dokl. 3 (1962), 707–710.
- M. G. Kreĭn, On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268–271 (Russian). MR 0139006
- B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088, DOI 10.1515/9783110941937
- 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, Introduction to spectral theory: selfadjoint ordinary differential operators, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein. MR 0369797, DOI 10.1090/mmono/039
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- 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
- H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
- H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143–226. MR 427731, DOI 10.1002/cpa.3160290203
- D. B. Pearson, Quantum scattering and spectral theory, Techniques of Physics, vol. 9, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1988. MR 1099604
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419 B. Simon, Spectral analysis of rank one perturbations and applications, Mathematical Quantum Theory II: Schrödinger Operators (J. Feldman, R. Froese, L.M. Rosen, eds.), Conf. Proc. Canad. Math. Soc., vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149.
- E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. MR 430403, DOI 10.1002/cpa.3160300305
- Stephanos Venakides, The infinite period limit of the inverse formalism for periodic potentials, Comm. Pure Appl. Math. 41 (1988), no. 1, 3–17. MR 917122, DOI 10.1002/cpa.3160410103
- J. Zorbas, Perturbation of self-adjoint operators by Dirac distributions, J. Math. Phys. 21 (1980), no. 4, 840–847. MR 565731, DOI 10.1063/1.524464
Bibliographic Information
- F. Gesztesy
- MR Author ID: 72880
- Email: mathfg@mizzou1.missouri.edu
- Received by editor(s): February 27, 1995
- Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The U.S. Government has certain rights in this material.
- © Copyright 1996 by the authors
- Journal: Trans. Amer. Math. Soc. 348 (1996), 349-373
- MSC (1991): Primary 34B24, 34L05, 81Q10; Secondary 34B20, 47A10
- DOI: https://doi.org/10.1090/S0002-9947-96-01525-5
- MathSciNet review: 1329533