On the double commutation method
Authors:
F. Gesztesy and G. Teschl
Journal:
Proc. Amer. Math. Soc. 124 (1996), 1831-1840
MSC (1991):
Primary 34B24, 34L05; Secondary 34B20, 47A10
DOI:
https://doi.org/10.1090/S0002-9939-96-03299-6
MathSciNet review:
1322925
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Abstract | References | Similar Articles | Additional Information
Abstract: We provide a complete spectral characterization of the double commutation method for general Sturm-Liouville operators which inserts any finite number of prescribed eigenvalues into spectral gaps of a given background operator. Moreover, we explicitly determine the transformation operator which links the background operator to its doubly commuted version (resulting in extensions and considerably simplified proofs of spectral results even for the special case of Schrödinger-type operators).
- 1. Tuncay Aktosun, Martin Klaus, and Cornelis van der Mee, Scattering and inverse scattering in one-dimensional nonhomogeneous media, J. Math. Phys. 33 (1992), no. 5, 1717–1744. MR 1158993, https://doi.org/10.1063/1.529650
- 2. Bernhard Baumgartner, Level comparison theorems, Ann. Physics 168 (1986), no. 2, 484–526. MR 844889, https://doi.org/10.1016/0003-4916(86)90041-2
- 3. R. F. Bikbaev and R. A. Sharipov, The asymptotic behavior, as 𝑡→∞, of the solution of the Cauchy problem for the Korteweg-de Vries equation in a class of potentials with finite-gap behavior as 𝑥→±∞, Teoret. Mat. Fiz. 78 (1989), no. 3, 345–356 (Russian, with English summary); English transl., Theoret. and Math. Phys. 78 (1989), no. 3, 244–252. MR 996221, https://doi.org/10.1007/BF01017661
- 4. M. M. Crum, Associated Sturm--Liouville systems, Quart. J. Math. Oxford (2) 6, 121--127 (1955). MR 17:266g
- 5. G. Darboux, Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. (Paris) 94, 1456--1459 (1882).
- 6. P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310. MR 495676
- 7. P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, https://doi.org/10.1002/cpa.3160320202
- 8. Michael S. P. Eastham and Hubert Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667015
- 9. M. S. P. Eastham and J. B. McLeod, The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators, Proc. Roy. Soc. Edinburgh Sect. A 79 (1977/78), no. 1-2, 25–34. MR 477250, https://doi.org/10.1017/S0308210500016796
- 10. Fritz Ehlers and Horst Knörrer, An algebro-geometric interpretation of the Bäcklund-transformation for the Korteweg-de Vries equation, Comment. Math. Helv. 57 (1982), no. 1, 1–10. MR 672841, https://doi.org/10.1007/BF02565842
- 11. N. M. Ercolani and H. Flaschka, The geometry of the Hill equation and of the Neumann system, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), no. 1533, 405–422. New developments in the theory and application of solitons. MR 836742, https://doi.org/10.1098/rsta.1985.0048
- 12. N. E. Firsova, Solution of the Cauchy problem for the Korteweg-de Vries equation with initial data that are the sum of a periodic and a rapidly decreasing function, Mat. Sb. (N.S.) 135(177) (1988), no. 2, 261–268, 272 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 1, 257–265. MR 937811, https://doi.org/10.1070/SM1989v063n01ABEH003272
- 13. Hermann Flaschka and David W. McLaughlin, Some comments on Bäcklund transformations, canonical transformations and the inverse scattering method, Bäcklund transformations, the inverse scattering method, solitons, and their applications (Workshop Contact Transformations, Vanderbilt Univ., Nashville, Tenn., 1974) Springer, Berlin, 1976, pp. 253–295. Lecture Notes in Math., Vol. 515. MR 0609534
- 14. F. R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsa, New York, 1990.
- 15. Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133. MR 336122, https://doi.org/10.1002/cpa.3160270108
- 16. I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser 2, 1, 253--304 (1955). MR 17:489c
- 17. F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), no. 2, 401–446. MR 1244942, https://doi.org/10.1006/jfan.1993.1132
- 18. F. Gesztesy and R. Svirsky, (m)KdV-Solitons on the background of quasi-periodic finite-gap solutions, Memoirs Amer. Math. Soc. 118, No. 563 (1995). CMP 95:04
- 19. F. Gesztesy and G. Teschl, Commutation methods for Jacobi operators, J. Differential Equations (to appear).
- 20. F. Gesztesy and K. Unterkofler, Isospectral deformations for Sturm-Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), no. 2, 113–137. MR 1227036, https://doi.org/10.1016/0034-4877(92)90008-O
- 21. F. Gesztesy and R. Weikard, Spectral deformations and soliton equations, Differential equations with applications to mathematical physics, Math. Sci. Engrg., vol. 192, Academic Press, Boston, MA, 1993, pp. 101–139. MR 1207152, https://doi.org/10.1016/S0076-5392(08)62376-0
- 22. F. Gesztesy and Z. Zhao, On critical and subcritical Sturm-Liouville operators, J. Funct. Anal. 98 (1991), no. 2, 311–345. MR 1111572, https://doi.org/10.1016/0022-1236(91)90081-F
- 23. F. Gesztesy, W. Schweiger, and B. Simon, Commutation methods applied to the mKdV-equation, Trans. Amer. Math. Soc. 324 (1991), no. 2, 465–525. MR 1029000, https://doi.org/10.1090/S0002-9947-1991-1029000-7
- 24. C. G. J. Jacobi, Zur Theorie der Variationsrechnung und der Differentialgleichungen, J. Reine Angew. Math. 17, 68--82 (1837).
- 25. I. Kay and H. E. Moses, Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys. 27, 1503--1508 (1956). MR 19:805e
- 26. I. M. Krichever, Potentials with zero coefficient of reflection on a background of finite--zone potentials, Funct. Anal. Appl. 9, 161--163 (1975).
- 27. E. A. Kuznetsov and A. V. Mikhaĭlov, Stability of stationary waves in nonlinear weakly dispersive media, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 5, 1717–1727 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 5, 855–859. MR 0387847
- 28. H. P. McKean, Geometry of KdV. I. Addition and the unimodular spectral classes, Rev. Mat. Iberoamericana 2 (1986), no. 3, 253–261. MR 908052
- 29. H. P. McKean, Geometry of KdV. II. Three examples, J. Statist. Phys. 46 (1987), no. 5-6, 1115–1143. MR 893135, https://doi.org/10.1007/BF01011159
- 30. B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
- 31. V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986.
- 32. Rafael R. del Río Castillo, Embedded eigenvalues of Sturm-Liouville operators, Comm. Math. Phys. 142 (1991), no. 2, 421–431. MR 1137069
- 33. A. V. Rybin and M. A. Sall′, Solitons of the Korteweg-de Vries equation on the background of a known solution, Teoret. Mat. Fiz. 63 (1985), no. 3, 333–339 (Russian, with English summary). MR 805515
- 34. U.-W. Schmincke, On Schrödinger’s factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 1-2, 67–84. MR 529570, https://doi.org/10.1017/S0308210500010143
- 35. R. A. Sharipov, Finite-gap analogues of 𝑁-multiplet solutions of the Korteweg-de Vries equation, Uspekhi Mat. Nauk 41 (1986), no. 5(251), 203–204 (Russian). MR 878353
- 36. R. A. Sharipov, Soliton multiplets of the Korteweg--de Vries equation, Sov. Phys. Dokl. 32, 121--123 (1987).
- 37. Christine Thurlow, The point-continuous spectrum of second-order, ordinary differential operators, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 197–211. MR 559666, https://doi.org/10.1017/S030821050001708X
- 38. Hugo D. Wahlquist, Bäcklund transformation of potentials of the Korteweg-de Vries equation and the interaction of solitons with cnoidal waves, Bäcklund transformations, the inverse scattering method, solitons, and their applications (Workshop Contact Transformations, Vanderbilt Univ., Nashville, Tenn., 1974) Springer, Berlin, 1976, pp. 162–183. Lecture Notes in Math., Vol. 515. MR 0610088
- 39. Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954
of the solution to the Cauchy problem for the Korteweg--de Vries equation in the class of potentials with finite--gap behavior as 
, Theoret. Math. Phys. 78, 244--252 (1989). Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34B24, 34L05, 34B20, 47A10
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Additional Information
F. Gesztesy
Affiliation:
Department of Mathematics, University of Missouri, Colum-bia, Missouri 65211
Email:
mathfg@mizzou1.missouri.edu
G. Teschl
Affiliation:
Department of Theoretical Physics, Technical University of Graz, Graz, 8010, Austria
Address at time of publication:
Institut für Reine und Angewandte Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen, D-52056 Aachen, Germany
Email:
mathgr42@mizzou1.missouri.edu, teschl@iram.rwth-aachen.de
DOI:
https://doi.org/10.1090/S0002-9939-96-03299-6
Keywords:
Commutation methods,
Sturm-Liouville operators,
eigenvalues
Received by editor(s):
December 8, 1994
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1996
American Mathematical Society
