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Hadamard’s problem and Coxeter groups: New examples of Huygens’ equations. (English. Russian original) Zbl 0845.35062
Funct. Anal. Appl. 28, No. 1, 3-12 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 3-15 (1994).
The main result of the work is the following. By a root system we mean a finite set $${\mathfrak R}= \{\alpha\}\subseteq \mathbb{R}^n$$ of vectors that have pairwise different directions, span $$\mathbb{R}^n$$, and satisfy the condition that the reflections $$s_\alpha(\beta)= \beta- 2\alpha(\alpha, \beta)/(\alpha, \alpha)$$ transform $$\mathfrak R$$ into itself. The reflections $$s_\alpha$$ generate a finite group $$\mathcal W$$ referred to as the Coxeter group. Let us define the potential associated with $$\mathcal R$$ by setting $u(x)= \sum_{\alpha\in {\mathfrak R}_+} {m_\alpha(m_\alpha+ 1)(\alpha, \alpha)\over (\alpha, x)^2},$ where $$m_\alpha$$ are integers such that the function $$m(\alpha)= m_\alpha$$ is $${\mathcal W}$$-invariant, and $${\mathfrak R}_+\subset {\mathfrak R}$$ denotes the set of roots positive with respect to a suitable linear form on $$\mathbb{R}^n$$. Consider the hyperbolic operator ${\mathcal L}= {\partial^2\over \partial t^2}- {\partial^2\over \partial x^2_1}-\cdots- {\partial^2\over \partial x^2_n}- {\partial^2\over \partial y^2_1}-\cdots- {\partial^2\over \partial y^2_m}+ u(x_1,\dots, x_n),$ where $$N= m+ n$$ is an arbitrary odd number satisfying the inequality $$N\geq 3+ 2 \sum_{\alpha\in {\mathfrak R}_+} m_\alpha$$. Theorem. For any Coxeter group $$\mathcal W$$ and integer-valued $${\mathcal W}$$-invariant function $$m$$ on the corresponding root system $$\mathfrak R$$ the equation $${\mathcal L}\varphi= 0$$ satisfies Huygens’ principle.

##### MSC:
 35L10 Second-order hyperbolic equations
##### Keywords:
Hadamard’s problem; Coxeter group; Huygens’ principle
Full Text:
##### References:
  I. G. Petrovsky, Lectures on Partial Differential Equations [in Russian], GITTL, Moscow?Leningrad (1950). · Zbl 0059.08402  R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, New York (1964). · Zbl 0121.07801  J. Hadamard, Lecture on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven (1923). · JFM 49.0725.04  M. Mathisson, ?Le problème de Hadamard relatif à la diffusion des ondes,? Acta Math.,71, 249-282 (1939). · Zbl 0022.22802  L. Asgeirsson, ?Some hints on Huygens’ principle and Hadamard’s conjecture,? Comm. Pure Appl. Math.,9, No. 3, 307-327 (1956). · Zbl 0074.31101  K. L. Stellmacher, ?Ein Beispiel einer Huygensschen Differentialgleichung,? Nachr. Akad. Wiss. Göttingen Math.?Phys. Kl. IIa,10, 133-138 (1953). · Zbl 0052.09901  J. E. Lagnese and K. L. Stellmacher, ?A method of generating classes of Huygens’ operators,? J. Math. Mech.,17, No. 5, 461-472 (1967). · Zbl 0154.36002  J. E. Lagnese, ?A solution of Hadamard’s problem for a restricted class of operators,? Proc. Amer. Math. Soc.,19, 981-988 (1968). · Zbl 0159.14203  G. Darboux, ?Sur la representations sphérique des surfaces,? Compt. Rend.,94, 1343-1345 (1882).  M. Adler and J. Moser, ?On a class of polynomials connected with the Korteweg?de Vries equation,? Commun. Math. Phys.,61, No. 1, 1-30 (1978). · Zbl 0428.35067  S. P. Novikov, Periodic problem for Korteweg?de Vries equation. I,? Funkts. Anal. Prilozhen.,8, No. 3, 54-63 (1974). · Zbl 0301.54027  Yu. Yu. Berest, ?Deformations preserving Huygens’ principle,? to appear in J. Math. Phys. (1993).  Yu. Yu. Berest and A. P. Veselov, ?Huygens’ principle and Coxeter groups,? Usp. Mat. Nauk,48, No. 3, 181-182 (1993).  M. A. Olshanetsky and A. M. Perelomov, ?Quantum completely integrable systems connected with semisimple Lie algebras,? Lett. Math. Phys.,2, 7-13 (1977). · Zbl 0366.58005  M. A. Olshanetsky and A. M. Perelomov, ?Quantum integrable systems related to Lie algebras,? Phys. Rep.,94, 313-404 (1983).  F. Calogero, ?Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potential,? J. Math. Phys.,12, 419-436 (1971).  O. A. Chalykh and A. P. Veselov, ?Commutative rings of partial differential operators and Lie algebras,? Preprint of FIM, ETH, Zürich (1988); Commun. Math. Phys.,126, 597-611 (1990). · Zbl 0746.47025  A. P. Veselov, K. L. Styrkas, and O. A. Chalykh, ?Algebraic integrability for the Schrödinger equation and the groups generated by reflections,? Teor. Mat. Fiz.,94, No 2, 253-275 (1993). · Zbl 0805.47070  E. M. Opdam, ?Some applications of hypergeometric shift operators,? Invent. Math.,98, 1-18 (1989). · Zbl 0696.33006  C. F. Dunkl, ?Differential-difference operators associated with reflection groups,? Trans. Amer. Math. Soc.,311, 167-183 (1989).  G. J. Heckman, ?A remark on the Dunkl differential-difference operators,? Progr. Math.,101, 181-191 (1991). · Zbl 0749.33005  G. Felder and A. P. Veselov, ?Shift operator for Calogero?Sutherland quantum problems via the Knizhnik?Zamolodchikov equation,? Preprint FIM, ETH, Zürich (1993); to appear in Commun. Math. Phys. · Zbl 0811.35105  V. N. Babich, ?Hadamard’s anzats: its analogs, generalizations, and applications,? Algebra Analiz,3, No. 5, 1-37 (1991). · Zbl 0779.35002  N. H. Ibragimov and A. O. Oganesyan, ?Hierarchy of Huygens’ equations in spaces with a nontrivial conformal group,? Usp. Mat. Nauk,46, No. 3, 111-146 (1991). · Zbl 0778.35067  J. Weiss, M. Tabor, and G. Carnevale, ?The Painlevé property for partial differential equations,? J. Math. Phys.,24, 522-526 (1983). · Zbl 0514.35083  J. Weiss, ?The sine-Gordon equation: complete and partial integrability,? J. Math. Phys.,25, 2226-2235 (1984). · Zbl 0557.35107  K. L. Stellmacher, ?Eine Klasse Huygensscher Differentialgleichungen und ihre Integration,? Math. Ann.,130, No. 3, 219-233 (1955). · Zbl 0134.31101  I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1, Acad. Press, New York (1964). · Zbl 0115.33101  I. N. Bernshtein, I. M. Gel’fand, and S. I. Gel’fand, ?Shubert cells and the cohomology ofG/P,? Usp. Mat. Nauk,28, No. 3, 3-26 (1973).
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