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TMF, 1993, Volume 94, Number 2, Pages 253–275 (Mi tmf1421)  

This article is cited in 16 scientific papers (total in 17 papers)

Algebraic integrability for the Schrödinger equation and finite reflection groups

A. P. Veselova, K. L. Styrkasb, O. A. Chalykhb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b M. V. Lomonosov Moscow State University

Abstract: Algebraic integrability of an $n$-dimensional Schrödinger equation means that it has more thann independent quantum integrals. For $n=1$, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero–Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero–Sutherland problem for a special value of the coupling constant.

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English version:
Theoretical and Mathematical Physics, 1993, 94:2, 182–197

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Received: 23.12.1992

Citation: A. P. Veselov, K. L. Styrkas, O. A. Chalykh, “Algebraic integrability for the Schrödinger equation and finite reflection groups”, TMF, 94:2 (1993), 253–275; Theoret. and Math. Phys., 94:2 (1993), 182–197

Citation in format AMSBIB
\by A.~P.~Veselov, K.~L.~Styrkas, O.~A.~Chalykh
\paper Algebraic integrability for the Schr\"odinger equation and finite reflection groups
\jour TMF
\yr 1993
\vol 94
\issue 2
\pages 253--275
\jour Theoret. and Math. Phys.
\yr 1993
\vol 94
\issue 2
\pages 182--197

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    This publication is cited in the following articles:
    1. Yu. Yu. Berest, A. P. Veselov, “The Huygens principle and Coxeter groups”, Russian Math. Surveys, 48:3 (1993), 183–184  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Yu. Yu. Berest, A. P. Veselov, “Hadamard's Problem and Coxeter Groups: New Examples of Huygens' Equations”, Funct. Anal. Appl., 28:1 (1994), 3–12  mathnet  crossref  mathscinet  zmath  isi
    3. Yu. Yu. Berest, A. P. Veselov, “Huygens' principle and integrability”, Russian Math. Surveys, 49:6 (1994), 5–77  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. P. Veselov, “Calogero quantum problem, Knizhnik–Zamolodchikov equation and Huygens principle”, Theoret. and Math. Phys., 98:3 (1994), 368–376  mathnet  crossref  mathscinet  zmath  isi
    5. A. P. Veselov, M. V. Feigin, O. A. Chalykh, “New integrable deformations of the Calogero–Moser quantum problem”, Russian Math. Surveys, 51:3 (1996), 573–574  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. O. A. Chalykh, “Additional integrals of the generalized quantum Calogero–Moser problem”, Theoret. and Math. Phys., 109:1 (1996), 1269–1273  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. O. A. Chalykh, “The duality of the generalized Calogero and Ruijsenaars problems”, Russian Math. Surveys, 52:6 (1997), 1289–1291  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. Yu. Yu. Berest, A. P. Veselov, “On the singularities of potentials of exactly soluble Schrödinger equations and on Hadamard's problem”, Russian Math. Surveys, 53:1 (1998), 208–209  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. Chalykh, O, “New integrable generalizations of Calogero–Moser quantum problem”, Journal of Mathematical Physics, 39:2 (1998), 695  crossref  mathscinet  zmath  adsnasa  isi
    10. Chalykh, OA, “Multidimensional integrable Schrodinger operators with matrix potential”, Journal of Mathematical Physics, 40:11 (1999), 5341  crossref  mathscinet  zmath  adsnasa  isi
    11. P. Etingof, V. A. Ginzburg, “On $m$-quasi-invariants of a Coxeter group”, Mosc. Math. J., 2:3 (2002), 555–566  mathnet  crossref  mathscinet  zmath
    12. Taniguchi, K, “On the symmetry of commuting differential operators with sinpularities along hyperplanes”, International Mathematics Research Notices, 2004, no. 36, 1845  crossref  mathscinet  zmath  isi
    13. Toshio Oshima, “Completely Integrable Systems Associated with Classical Root Systems”, SIGMA, 3 (2007), 061, 50 pp.  mathnet  crossref  mathscinet  zmath
    14. Chalykh, O, “Algebro-geometric Schrodinger operators in many dimensions”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 947  crossref  mathscinet  zmath  adsnasa  isi
    15. A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. V. E. Adler, Yu. Yu. Berest, V. M. Buchstaber, P. G. Grinevich, B. A. Dubrovin, I. M. Krichever, S. P. Novikov, A. N. Sergeev, M. V. Feigin, J. Felder, E. V. Ferapontov, O. A. Chalykh, P. I. Etingof, “Alexander Petrovich Veselov (on his 60th birthday)”, Russian Math. Surveys, 71:6 (2016), 1159–1176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154  mathnet  crossref  crossref  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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