Singular Eigenfunctions of Calogero–Sutherland Type Systems and How to Transform Them into Regular Ones

Edwin Langmann

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

There exists a large class of quantum many-body systems of Calogero–Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable $\mathcal{PT}$-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero–Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones.

Détail

Keywords: quantum integrable systems; orthogonal polynomials; singular eigenfunctions.

Bibliographie

[1] Calogero F., “Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys., 12 (1971), 419–436 | DOI | MR

[2] Sutherland B., “Exact results for a quantum many-body problem in one dimension. II”, Phys. Rev. A, 5:3 (1972), 1372–1376 | DOI | MR

[3] Kuznetsov V. B., Mangazeev V. V., Sklyanin E. K., “$Q$-operator and factorised separation chain for Jack polynomials”, Indag. Math. (N.S.), 14 (2003), 451–482 ; math.CA/0306242 | DOI | MR | Zbl

[4] Langmann E., “Anyons and the elliptic Calogero–Sutherland model”, Lett. Math. Phys., 54 (2000), 279–289 ; math-ph/0007036 | DOI | MR | Zbl

[5] Langmann E., “Algorithms to solve the (quantum) Sutherland model”, J. Math. Phys., 42 (2001), 4148–4157 ; math-ph/0104039 | DOI | MR | Zbl

[6] Langmann E., An explicit solution of the (quantum) elliptic Calogero–Sutherland model, math-ph/0407050 | MR

[7] Langmann E., “A method to derive explicit formulas for an elliptic generalization of the Jack polynomials”, Proceedings of Workshop “Jack, Hall–Littlewood and Macdonald polynomials” (September 23–26, 2003, Edinburgh), Contemp. Math., 417, eds. V. B. Kuznetsov and S. Sahi, 2006, 257–270 ; math-ph/0511015 | MR | Zbl

[8] Langmann E., An algorithm to solve the elliptic Calogero–Sutherland model, math-ph/0401029 | MR

[9] Kuznetsov V. B., Sklyanin E. K., “On Bäcklund transformations for many-body systems”, J. Phys. A: Math. Gen., 31 (1998), 2241–2251 ; solv-int/9711010 | DOI | MR | Zbl

[10] Kuznetsov V. B., Petrera M., Ragnisco O., “Separation of variables and Bäcklund transformations for the symmetric Lagrange top”, J. Phys. A: Math. Gen., 37 (2004), 8495–8512 ; nlin.SI/0403028 | DOI | MR | Zbl

[11] Bender C. M., Boettcher S., Meisinger P., “PT-symmetric quantum mechanics”, J. Math. Phys., 40 (1999), 2201–2229 ; quant-ph/9809072 | DOI | MR | Zbl

[12] Stanley R. P., “Some combinatorial properties of Jack symmetric functions”, Adv. Math., 77 (1989), 76–115 | DOI | MR | Zbl

[13] Dunkl C. F., Xu Y., Orthogonal polynomials of several variables, Cambridge University Press, Cambridge, 2001 | MR | Zbl

[14] Hallnäs M., Langmann E., “Explicit formulas for the eigenfunctions of the $N$-body Calogero model”, J. Phys. A: Math. Gen., 39 (2006), 3511–3533 ; math-ph/0511040 | DOI | MR | Zbl

[15] Hallnäs M., Langmann E., Quantum Calogero–Sutherland type models associtated with orthogonal polynomials, work in progress

[16] Sen D., “A multispecies Calogero–Sutherland model”, Nuclear Phys. B, 479 (1996), 554–574 ; cond-mat/9512014 | DOI | MR | Zbl

[17] Heckman G. J., Opdam E. M., “Root systems and hypergeometric functions. I”, Compos. Math., 64 (1987), 329–352 | MR | Zbl

[18] Chalykh O., Feigin M., Veselov A., “New integrable generalizations of Calogero–Moser quantum problem”, J. Math. Phys., 39 (1998), 695–703 | DOI | MR | Zbl

[19] Sergeev A. N., Veselov A. P., “Deformed quantum Calogero–Moser problems and Lie superalgebras”, Comm. Math. Phys., 245 (2005), 249–278 ; math-ph/0303025 | DOI | MR

[20] Sergeev A. N., Veselov A. P., “Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials”, Adv. Math., 192 (2005), 341–375 ; math-ph/0307036 | DOI | MR | Zbl

[21] Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975 | MR | Zbl

[22] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1979 | MR | Zbl

[23] Awata H., Fukuma M. Matsuo Y., Odake S., “Representation theory of the $W_{1+\infty}$ algebra”, Progr. Theoret. Phys. Suppl., 118 (1995), 343–374 ; hep-th/9408158 | DOI | MR

[24] Mimachi K., Yamada Y., “Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials”, Comm. Math. Phys., 174 (1995), 447–455 | DOI | MR | Zbl

[25] Awata H., Matsuo Y., Odake S., Shiraishi J., “Excited states of Calogero–Sutherland model and singular vectors of the $W_N$ algebra”, Nuclear Phys. B, 449 (1995), 347–374 ; hep-th/9503043 | DOI | MR

[26] Wojciechowski S., “The analogue of the Bäcklund transformation for integrable many-body systems”, J. Phys. A: Math. Gen., 15 (1982), L653–L657 | DOI | MR