Abstract
We show how recent developments in the theory of (quantum) integrable systems can be applied to the study of lacunas of hyperbolic equations, one of the classical problems in analysis of linear differential operators. This report is based mostly on results of our recent work [3].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. F. Atiyah, R. Bott, and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109–189.
M. F. Atiyah, R. Bott, and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math. 131 (1973), 145–206.
Yu. Yu. Berest, The problem of lacunae and analysis on root systems, Tech. Report CRM-2468, Centre de recherches mathématiques, 1997.
Yu. Yu. Berest and A. P. Veselov, The Hadamard problem and Coxeter groups: new examples of the Huygens equations, Funct. Anal. Appl. 28 (1994), No. 1, 3–12.
C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), No. 1, 167–183.
L. Gårding,Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85 (1951), 1–62.
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, CT, 1923.
G. J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic Analysis on Reductive Groups (Bowdoin, 1989) (W. Barker and P. Sally, eds.), Progr. Math., Vol. 101, Birkhäuser, Boston, MA, 1991, pp. 181–191.
L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Grundlehren Math. Wiss., Vol. 256, Springer, New York, 1983.
L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients, Grundlehren Math. Wiss., Vol. 257, Springer, New York, 1983.
L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators, Grundlehren Math. Wiss., Vol. 274, Springer, New York, 1985.
L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Grundlehren Math. Wiss., Vol. 275, Springer, New York, 1985.
M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), No. 6, 313–404.
I. G. Petrowsky, On the diffusion of waves and the lacunas for hyperbolic equations, Mat. Sbornik 17 (1945), 289–370.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
Berest, Y.Y. (2000). The Theory of Lacunas and Quantum Integrable Systems. In: van Diejen, J.F., Vinet, L. (eds) Calogero—Moser— Sutherland Models. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1206-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1206-5_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7043-0
Online ISBN: 978-1-4612-1206-5
eBook Packages: Springer Book Archive