Algebraic integrability of Schrodinger operators and representations of Lie algebras

Pavel Etingof; Konstantin Styrkas

Compositio Mathematica (1995)

  • Volume: 98, Issue: 1, page 91-112
  • ISSN: 0010-437X

How to cite

top

Etingof, Pavel, and Styrkas, Konstantin. "Algebraic integrability of Schrodinger operators and representations of Lie algebras." Compositio Mathematica 98.1 (1995): 91-112. <http://eudml.org/doc/90396>.

@article{Etingof1995,
author = {Etingof, Pavel, Styrkas, Konstantin},
journal = {Compositio Mathematica},
keywords = {highest weight modules; Calogero-Sutherland operator; algebraic integrability; matrix Schrödinger operators; complex simple Lie algebra},
language = {eng},
number = {1},
pages = {91-112},
publisher = {Kluwer Academic Publishers},
title = {Algebraic integrability of Schrodinger operators and representations of Lie algebras},
url = {http://eudml.org/doc/90396},
volume = {98},
year = {1995},
}

TY - JOUR
AU - Etingof, Pavel
AU - Styrkas, Konstantin
TI - Algebraic integrability of Schrodinger operators and representations of Lie algebras
JO - Compositio Mathematica
PY - 1995
PB - Kluwer Academic Publishers
VL - 98
IS - 1
SP - 91
EP - 112
LA - eng
KW - highest weight modules; Calogero-Sutherland operator; algebraic integrability; matrix Schrödinger operators; complex simple Lie algebra
UR - http://eudml.org/doc/90396
ER -

References

top
  1. [C] Calogero, F.: Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys.12 (1971), 419-436. Zbl1002.70558MR280103
  2. [Ch] Cherednik, I.: Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation, preprint; Submitted to Comm. Math. Phys. (1994). Zbl0826.35100MR1329203
  3. [CV1] Chalykh, O.A. and Veselov, A.P.: Integrability in the theory of Schrödinger operator and harmonic analysis, Comm. Math. Phys.152 (1993), 29-40. Zbl0767.35066MR1207668
  4. [CV2] Chalykh, O.A. and Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys.126 (1990), 597-611. Zbl0746.47025MR1032875
  5. [E] Etingof, P.I.: Quantum integrable systems and representations of Lie algebras, hep-th 9311132, Journal of Mathematical Physics (1993), to appear. Zbl0861.17002MR1331279
  6. [EK1] Etingof, P.I. and Kirillov Jr., A.A.: Representations of affine Lie algebras, parabolic differential equations, and Lamé functions, hep-th 9310083, Duke Math. J., vol. 74(3), 1994, pages 585-614. Zbl0811.17026MR1277946
  7. [EK2] Etingof, P.I. and Kirillov,, Jr. A.A.: A unified representation-theoretic approach to special functions, hep-th 9312101, Functional Anal. and its Applic.28 (1994), no. 1. Zbl0868.33010
  8. [G] Grinevich, P.G.: Commuting matrix differential operators of arbitrary rank, Soviet Math. Dokl.30 (1984), no. 2, 515-518. Zbl0598.47049MR765610
  9. [HO] Heckman, G.J. and Opdam, E.M.: Root systems and hypergeometric functions I, Compos. Math.64 (1987), 329-352. Zbl0656.17006MR918416
  10. [H1] Heckman, G.J.: Root systems and hypergeometric functions II, Compos. Math.64 (1987), 353-373. Zbl0656.17007MR918417
  11. [KK] Kac, V.G. and Kazhdan, D.A.: Structure of representations with highest weight of infinite dimensional Lie algebras, Advances in Math.34 (1979), no. 1, 97-108. Zbl0427.17011MR547842
  12. [Kr] Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surv.32:6 (1977), 185-213. Zbl0386.35002
  13. [O1] Opdam, E.M.: Root systems and hypergeometric functions III, Compos. Math.67 (1988), 21-49. Zbl0669.33007MR949270
  14. [O2] Opdam, E.M.: Root systems and hypergeometric functions IV, Compos. Math.67 (1988), 191-207. Zbl0669.33008MR951750
  15. [OOS] Ochiai, H., Oshima, T. and Sekiguchi, H.: Commuting families of symmetric differential operators, (preprint), Univ. of Tokyo (1994). Zbl0817.22010MR1272672
  16. [OP] Olshanetsky, M.A. and Perelomov, A.M.: Quantum integrable systems related to Lie algebras, Phys. Rep.94 (1983), 313-404. Zbl0366.58005MR708017
  17. [Osh] Oshima, T.: Completely integrable systems with a symmetry in coordinates, (preprint), Univ. of Tokyo (1994). Zbl0965.37055MR1734134
  18. [OS] Oshima, T. and Sekiguchi, H.: Commuting families of differential operators invariant under the action of the Weyl group, (preprint) UTMS 93-43, Dept. of Math. Sci., Univ. of Tokyo (1993). Zbl0863.43007
  19. [S] Sutherland, B.: Exact results for a quantum many-body problem in one dimension, Phys. Rev.A5 (1972), 1372-1376. 
  20. [Sh] Shapovalov, N.N.: On bilinear form on the universal enveloping algebra of a simple Lie algebra, Funct. Anal. Appl.6 (1972), 307-312. Zbl0283.17001
  21. [VSC] Veselov, A.P., Styrkas, K.A. and Chalykh, O.A.: Algebraic integrability for the Schrödinger equation and finite reflection groups, Theor. and Math. Physics94 (1993), no. 2. Zbl0805.47070MR1221735

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.