Explicit Solution of the (Quantum) Elliptic Calogero-Sutherland Model

The elliptic Calogero–Sutherland model is a quantum many body system of identical particles moving on a circle and interacting via two body potentials proportional to the Weierstrass

{\wp}

-function. It also provides a natural many-variable generalization of the Lamé equation. Explicit formulas for the eigenfunctions and eigenvalues of this model as infinite series are obtained, to all orders and for arbitrary particle numbers and coupling parameters. These eigenfunctions are an elliptic deformation of the Jack polynomials. The absolute convergence of these series is proved in special cases, including the two-particle (=Lamé) case for non-integer coupling parameters and sufficiently small elliptic deformation.

Note:

v1: 17 pages. The solution is given as series in q but only to low order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper completely revised: the results of v1 and v2 are extended to all orders

References(48)

Figures(0)

Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials

F. Calogero
(
- Moscow, ITEP
)

- J.Math.Phys. 12 (1971) 419-436
•
DOI:
- 10.1063/1.1665604

Exact results for a quantum many body problem in one-dimension. 2.

Bill Sutherland
(
- Utah U.
)

- Phys.Rev.A 5 (1972) 1372-1376
•
DOI:
- 10.1103/PhysRevA.5.1372

Exactly Solvable One-Dimensional Many Body Problems

F. Calogero
(
- Moscow, ITEP
)

- Lett.Nuovo Cim. 13 (1975) 411
•
DOI:
- 10.1007/BF02790495

Quantum Completely Integrable Systems Connected with Semisimple Lie Algebras

M.A. Olshanetsky
(
- Moscow, ITEP
)
,
A.M. Perelomov
(
- Moscow, ITEP
)

- Lett.Math.Phys. 2 (1977) 7-13,
- Letters Math.Phys.2 (1977) 7-13 and Moscow Inst Theor Exp Ph Gkae - ITEF-172 (76,REC.AUG 77) 13p
•
DOI:
- 10.1007/BF00420664

Solution algorithm for the elliptic Calogero-Sutherland model

Edwin Langmann
(
- Royal Inst. Tech., Stockholm
)

- Lett.Math.Phys. 54 (2000) 279-289
•
e-Print:
- math-ph/0007036
•
DOI:
- 10.1023/A:1010961107811

Second quantization of the elliptic Calogero-Sutherland model

Edwin Langmann
(
- Royal Inst. Tech., Stockholm
)

- Commun.Math.Phys. 247 (2004) 321-351
•
e-Print:
- math-ph/0102005
•
DOI:
- 10.1007/s00220-004-1077-8

Loop groups, anyons and the Calogero-Sutherland model

- Commun.Math.Phys. 201 (1999) 1-34
•
e-Print:
- math-ph/9805010
•
DOI:
- 10.1007/s002200050547

The Perturbation of the quantum Calogero-Moser-Sutherland system and related results

Yasushi Komori
(
- Tokyo U., Komaba
)
,
Kouichi Takemura
(
- Yokohama City U.
)

- Commun.Math.Phys. 227 (2002) 93-118
•
e-Print:
- math/0009244
•
DOI:
- 10.1007/s002200200622

: Methods of Modern Mathematical Physics, vol. II Press, London

Reed M.
,
Simon B.

: Symmetric functions and Hall polynomials. In: Oxford Mathematical Monographs Press, Oxford

I.G. Macdonald

: Some properties of Jack symmetric functions

Stanley R.P.

- Adv.Math. 77 (1989) 76

Exact ground state wave function for a one-dimensional plasma

Bill Sutherland
(
- SUNY, Stony Brook
)

- Phys.Rev.Lett. 34 (1975) 1083
•
DOI:
- 10.1103/PhysRevLett.34.1083

An Explicit solution of the (quantum) elliptic Calogero-Sutherland model

Edwin Langmann
(
- Royal Inst. Tech., Stockholm
)

e-Print:
- math-ph/0407050
•
DOI:
- 10.1142/9789812702142_0020

: Algorithms to solve the (quantum) Sutherland model

Langmann E.

- J.Math.Phys. 42 (2001) 4148

Explicit Solution of the (Quantum) Elliptic Calogero-Sutherland Model

Edwin Langmann

- Annales Henri Poincare 15 (2014) 4, 755-791
•
e-Print:
- math-ph/0401029
•
DOI:
- 10.1007/s00023-013-0254-8

A method to derive explicit formulas for an elliptic generalization of the Jack polynomials

Edwin Langmann

- Jack, Hall-Littlewood and Macdonald polynomials, 257-270. Contemp. Math., 417 AMS (2006)
•
e-Print:
- math-ph/0511015

: Elliptic integrable systems of Calogero-Moser type: a survey. In: Noumi, M., Takasaki, K. (eds.) Proceedings of theKyoto Workshop on Elliptic integrable systems, p. 201. Rokko Lectures in Mathematics, Department of Mathematics, Kobe University (2005)

Y. Komori
,
S.N.M. Ruijsenaars

: Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge

E.T. Whittaker
,
G.N. Watson

Representation of affine Lie algebras, parabolic differential equations and Lame functions

- Duke Math.J. 74 (1994) 585-614
•
e-Print:
- hep-th/9310083
•
DOI:
- 10.1215/S0012-7094-94-07421-8

: Generalized Lamé functions. I. The elliptic case

Ruijsenaars S.N.M.

- J.Math.Phys. 40 (1999) 1595

Exact solutions of an elliptic Calogero-Sutherland model

- Phys.Lett.B 511 (2001) 112-118
•
e-Print:
- hep-th/0006039
•
DOI:
- 10.1016/S0370-2693(01)00573-1

: Lax representation with spectral parameter on a torus for integrable particle systems

Inozemtsev V.I.

- Lett.Math.Phys. 17 (1989) 11

: On the structure of eigenvectors of the multidimensional Lamé operator

Dittrich J.
,
Inozemtsev V.I.

- J.Phys.A 26 (1993) L753

: Solution to three-magnon problem for S = 1/2 periodic quantum spin chains with elliptic exchange

Inozemtsev V.I.

- J.Math.Phys. 37 (1996) 147

: The eigenvectors of the Heisenberg Hamiltonian with elliptic form of the exchange spin interaction

Inozemtsev V.I.

- J.Nonlin.Math.Phys. 12 (2005) 395

1-25 of 48
1
2
25 / page