Abstract
We present an approach to higher-dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct nontrivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero–Moser.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Andrianov, A., Borisov, N. and Ioffe, M.: The factorization method and quantum systems with equivalent energy spectra, Phys. Lett. A 105(1,2) (1984), 19–22.
Bakalov, B., Horozov, E. and Yakimov, M.: General methods for constructing bispectral operators Phys. Lett. A 222 (1996), 59–66.
Bakalov, B., Horozov, E. and Yakimov, M.: Bäcklund-Darboux transformations in Sato's Grassmannian, Serdica Math. J. 22 (1996), 571–588.
Bakalov, B., Horozov, E. and Yakimov, M.: Bispectral commutative rings of ordinary differential operators, Comm. Math. Phys. 190 (1997), 331–373.
Berest, Yu. and Kasman, A.: D-modules and Darboux transformations, Lett. Math. Phys. 43 (1998), 279–294.
Braverman, A., Etingof, P. and Gaitsgory, D.: Quantum integrable systems and differential Galois theory, Transform. Groups 2 (1997), 31–56.
Chalykh, O. A. and Veselov, A. P.: Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 125 (1990), 597–611.
Darboux, G.: Sur une proposition relative aux equation lineaires, C. R. Acad. Sci. Paris SérI Math 94 (1882), 1456.
Darboux, G.: Leçons sur la théorie géné rale des surfaces, 2éme partie, Paris, Gauthiers-Villars, 1889.
Duistermaat, J. J. and Grünbaum, F. A.: Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240.
Grünbaum, F. A.: Bispectral Musings, CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, RI, 1998, pp. 31–46.
Gonzalez-Lopez, A. and Kamran, N.: The multidimensional Darboux transformation, hep-th/961200.
Kasman, A. and Rothstein, M.: Bispectral Darboux transformations: The generalized Airy case, Physica D 102 (1997), 159–173.
Kasman, A.: Darboux transformations from n-KdV to KP, Acta Appl. Math. 49(2) (1997), 179–197.
Kasman, A.: Darboux Transformations and the Bispectral Problem, Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998, pp. 81–91.
Liberati, J.: Bispectral property, Darboux transformation and the Grassmannian Grrat, Lett. Math. Phys. 41(4) (1997), 321–332.
Matveev, V. B.: Darboux transformation and explicit solutions of the Kadomtsev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys. 3 (1979), 213–216.
Matveev, V. B. and Salle, M. A.: Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991.
Sabatier, P. On multidimensional Darboux transformations, Inverse Problems 14(2) (1998), 355–366.
Veselov, A. P.: Baker-Akhiezer Functions and the Bispectral Problem in Many Dimensions, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 1998, pp. 123–127.
Wilson, G.: Bispectral commutative ordinary differential operators, J. reine angew. Math. 442 (1993), 177–204.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Horozov, E., Kasman, A. Darboux Transformations of Bispectral Quantum Integrable Systems. Letters in Mathematical Physics 49, 131–143 (1999). https://doi.org/10.1023/A:1007618601784
Issue Date:
DOI: https://doi.org/10.1023/A:1007618601784