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The problem of lacunas and analysis on root systems

Author: Yuri Berest
Journal: Trans. Amer. Math. Soc. 352 (2000), 3743-3776
MSC (1991): Primary 58F07, 35L15; Secondary 58F37, 35L25
Published electronically: April 18, 2000
MathSciNet review: 1694280
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A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

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  • 1. Leifur Ásgeirsson, Some hints on Huygens’ principle and Hadamard’s conjecture, Comm. Pure Appl. Math. 9 (1956), 307–326. MR 0082034
  • 2. M. F. Atiyah, R. Bott, and L. Gȧrding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109–189. MR 0470499
  • 3. M. F. Atiyah, R. Bott, and L. Gȧrding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math. 131 (1973), 145–206. MR 0470500
  • 4. Yuri Berest, Lacunae of hyperbolic Riesz kernels and commutative rings of partial differential operators, Lett. Math. Phys. 41 (1997), no. 3, 227–235. MR 1463872, 10.1023/A:1007366720401
  • 5. Yuri Yu. Berest, Solution of a restricted Hadamard problem on Minkowski spaces, Comm. Pure Appl. Math. 50 (1997), no. 10, 1019–1052. MR 1466585, 10.1002/(SICI)1097-0312(199710)50:10<1019::AID-CPA3>3.0.CO;2-F
  • 6. Yu.Yu. Berest, The theory of lacunas and quantum integrable systems, in Proceedings of the Workshop The Calogero-Moser-Sutherland Model (J.-F. van Diejen and L. Vinet, Eds.), CRM Series in Mathematical Physics, Springer-Verlag (1998), to appear.
  • 7. Yuri Berest and Yuri Molchanov, Fundamental solutions for partial differential equations with reflection group invariance, J. Math. Phys. 36 (1995), no. 8, 4324–4339. MR 1341994, 10.1063/1.530964
  • 8. Yu. Yu. Berest and A. P. Veselov, The Huygens principle and Coxeter groups, Uspekhi Mat. Nauk 48 (1993), no. 3(291), 181–182 (Russian); English transl., Russian Math. Surveys 48 (1993), no. 3, 183–184. MR 1243616, 10.1070/RM1993v048n03ABEH001036
  • 9. Yu. Yu. Berest and A. P. Veselov, The Hadamard problem and Coxeter groups: new examples of the Huygens equations, Funktsional. Anal. i Prilozhen. 28 (1994), no. 1, 3–15, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 28 (1994), no. 1, 3–12. MR 1275723, 10.1007/BF01079005
  • 10. Yu. Yu. Berest and A. P. Veselov, The Huygens principle and integrability, Uspekhi Mat. Nauk 49 (1994), no. 6(300), 7–78 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 6, 5–77. MR 1316866, 10.1070/RM1994v049n06ABEH002447
  • 11. V. A. Borovikov, Some sufficient conditions for the absence of gaps, Mat. Sb. (N.S.) 55 (97) (1961), 237–254 (Russian). MR 0141887
  • 12. Louis Boutet de Monvel, Lacunas and transmissions, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 209–218. MR 547020
  • 13. J. Chaillou, Les polynômes différentiels hyperboliques et leurs perturbations singulières, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1973 (French). Avec une annexe de Christiane Rondeaux; Préface de J. Leray; Collection: “Varia Mathematica”. MR 0463709
  • 14. O. A. Chalykh and A. P. Veselov, Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990), no. 3, 597–611. MR 1032875
  • 15. Oleg A. Chalykh and Alexander P. Veselov, Integrability in the theory of Schrödinger operator and harmonic analysis, Comm. Math. Phys. 152 (1993), no. 1, 29–40. MR 1207668
  • 16. T. W. Chaundy, Hypergeometric partial differential equations (III), Quart. J. Math., Oxford Ser. 10 (1939), 219–240. MR 0000723
  • 17. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962. MR 0140802
  • 18. G. F. D. Duff, Singularities, supports and lacunas, Advances in microlocal analysis (Lucca, 1985) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 168, Reidel, Dordrecht, 1986, pp. 73–133. MR 850499
  • 19. J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183–269. MR 0388464
  • 20. Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. MR 951883, 10.1090/S0002-9947-1989-0951883-8
  • 21. Charles F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123–138. MR 1199124, 10.1090/conm/138/1199124
  • 22. F. G. Friedlander, The wave equation on a curved space-time, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Cambridge Monographs on Mathematical Physics, No. 2. MR 0460898
  • 23. I. G. Petrovskiĭ, Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Sistemy uravnenii s chastnymi proizvodnymi. Algebraicheskaya geometriya. [Systems of partial differential equations. Algebraic geometry]; Edited and with a preface by V. I. Arnol′d, N. N. Bogolyubov, A. N. Kolmogorov, O. A. Oleĭnik, S. L. Sobolev and A. N. Tikhonov; Compiled by Oleĭnik; With commentaries by Kolmogorov, L. R. Volevich, V. Ya. Ivriĭ, I. M. Gel′fand, G. E. Shilov, Oleĭnik, V. P. Palamodov, A. M. Gabrièlov and V. M. Kharlamov. MR 871873
  • 24. S. A. Gal′pern and V. E. Kondrašov, The Cauchy problem for differential operators which decompose into wave factors, Trudy Moskov. Mat. Obšč. 16 (1967), 109–136 (Russian). MR 0224986
  • 25. Lars Gårding, The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. of Math. (2) 48 (1947), 785–826. MR 0022648
  • 26. Lars Gårding, Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85 (1951), 1–62. MR 0041336
  • 27. Lars Gȧrding, Sharp fronts of paired oscillatory integrals, Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976), 1976/77 suppl, pp. 53–68. MR 0470501
    Lars Gȧrding, Corrections to: “Sharp fronts of paired oscillatory integrals” (Publ. Res. Inst. Math. Sci. 12 (1976/77), suppl., 53–68), Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 3, 821. MR 0470502
  • 28. Lars Gårding, Singularities in linear wave propagation, Lecture Notes in Mathematics, vol. 1241, Springer-Verlag, Berlin, 1987. MR 905058
  • 29. I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Translated by Eugene Saletan, Academic Press, New York-London, 1964. MR 0166596
  • 30. S. G. Gindikin, Analysis in homogeneous domains, Uspehi Mat. Nauk 19 (1964), no. 4 (118), 3–92 (Russian). MR 0171941
  • 31. Simon Gindikin, Tube domains and the Cauchy problem, Translations of Mathematical Monographs, vol. 111, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by Senya Shlosman. MR 1192129
  • 32. Paul Günther, Huygens’ principle and hyperbolic equations, Perspectives in Mathematics, vol. 5, Academic Press, Inc., Boston, MA, 1988. With appendices by V. Wünsch. MR 946226
  • 33. Jacques Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover Publications, New York, 1953. MR 0051411
  • 34. J. Hadamard, The problem of diffusion of waves, Ann. of Math. (2) 43 (1942), 510–522. MR 0006809
  • 35. Gerrit J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 181–191. MR 1168482
  • 36. Gerrit Heckman and Henrik Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16, Academic Press, Inc., San Diego, CA, 1994. MR 1313912
  • 37. Sigurdur Helgason, Wave equations on homogeneous spaces, Lie group representations, III (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1077, Springer, Berlin, 1984, pp. 254–287. MR 765556, 10.1007/BFb0072341
  • 38. Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 0388463
  • 39. Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
    Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278
    Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
    Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. MR 781537
  • 40. I. M. Kričever, Methods of algebraic geometry in the theory of nonlinear equations, Uspehi Mat. Nauk 32 (1977), no. 6(198), 183–208, 287 (Russian). MR 0516323
  • 41. John E. Lagnese, A solution of Hadamard’s problem for a restricted class of operators, Proc. Amer. Math. Soc. 19 (1968), 981–988. MR 0231024, 10.1090/S0002-9939-1968-0231024-7
  • 42. Anneli Lax, On Cauchy’s problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math. 9 (1956), 135–169. MR 0081406
  • 43. Hans Lewy, The wave equation as limit of hyperbolic equations of higher order, Comm. Pure Appl. Math. 18 (1965), 5–16. MR 0174868
  • 44. Myron Mathisson, Le problème de M. Hadamard relatif à la diffusion des ondes, Acta Math. 71 (1939), 249–282 (French). MR 0000728
  • 45. R. G. McLenaghan, Huygens’ principle, Ann. Inst. H. Poincaré Sect. A (N.S.) 37 (1982), no. 3, 211–236 (1983) (English, with French summary). MR 694586
  • 46. Wim Nuij, A note on hyperbolic polynomials, Math. Scand. 23 (1968), 69–72 (1969). MR 0250128
  • 47. M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), no. 6, 313–404. MR 708017, 10.1016/0370-1573(83)90018-2
  • 48. Philip Feinsilver and Jerzy Kocik, Representations of the Heisenberg algebra, Riccati systems, and associated geometric structures, Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (Clausthal, 1986) World Sci. Publ., Teaneck, NJ, 1987, pp. 255–262. MR 1023201
    E. M. Opdam, Root systems and hypergeometric functions. IV, Compositio Math. 67 (1988), no. 2, 191–209. MR 951750
  • 49. E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333–373. MR 1214452
  • 50. I. Petrowsky, On the diffusion of waves and the lacunas for hyperbolic equations, Rec. Math. [Mat. Sbornik] N. S. 17(59) (1945), 289–370 (English, with Russian summary). MR 0016861
  • 51. Marcel Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1–223 (French). MR 0030102
  • 52. O. G. Owens, A boundary-value problem for analytic solutions of an ultrahyperbolic equation, Duke Math. J. 21 (1954), 29–38. MR 0060697
  • 53. Karl L. Stellmacher, Eine Klasse huyghenscher Differentialgleichungen und ihre Integration, Math. Ann. 130 (1955), 219–233 (German). MR 0073831
  • 54. S. Leif Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat. 8 (1969), 145–162. MR 0271538
  • 55. B. R. Vaĭnberg and S. G. Gindikin, A strengthened Huygens principle for a certain class of differential operators with constant coefficients, Trudy Moskov. Mat. Obšč. 16 (1967), 151–180 (Russian). MR 0227592
  • 56. V. A. Vasil′ev, Sharpness and the local Petrovskiĭ condition for strictly hyperbolic operators with constant coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 2, 242–283 (Russian). MR 842583
  • 57. V. A. Vassiliev, Ramified integrals, singularities and lacunas, Mathematics and its Applications, vol. 315, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1336145
  • 58. A. P. Veselov, K. L. Styrkas, and O. A. Chalykh, Algebraic integrability for the Schrödinger equation, and groups generated by reflections, Teoret. Mat. Fiz. 94 (1993), no. 2, 253–275 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 94 (1993), no. 2, 182–197. MR 1221735, 10.1007/BF01019330
  • 59. A. P. Veselov, M. V. Feĭgin, and O. A. Chalykh, New integrable deformations of the quantum Calogero-Moser problem, Uspekhi Mat. Nauk 51 (1996), no. 3(309), 185–186 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 3, 573–574. MR 1406057, 10.1070/RM1996v051n03ABEH002956

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Additional Information

Yuri Berest
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201

Keywords: Hyperbolic linear differential operators, fundamental solution, lacuna, Huygens' principle, Coxeter groups, Dunkl operators
Received by editor(s): March 6, 1997
Received by editor(s) in revised form: November 3, 1997
Published electronically: April 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society