Quantum Integrable Systems and Special Functions

In classical mechanics, there is a well defined meaning to the term complete integrability. If, in n degrees of freedom, we have n independent functions in involution (mutually Poisson commuting), then the system can be integrated, ‘up to quadrature’. This is known as Liouville’s Theorem. In n-dimensions, quantum integrable systems are defined analogously by requiring the existence of n mutually commuting differential operators. However, we don’t have a theorem analogous to Liouville’s and we don’t have anything resembling ‘reduction to quadrature’. What we would like to achieve in quantum mechanics is to build the spectrum and corresponding eigenfunctions for Schrödinger’s equation with some given boundary conditions. This class of Schrödinger equation is called exactly solvable, which are rare, even in 1-dimension (where they are well documented). Separation of variables reduces the problem of solving one Schrödinger equation in n-dimensions to that of solving n Schrödinger equations in 1-dimension, but rarely results in exactly solvable cases.

One of the most powerful techniques for constructing exactly solvable potentials is the Darboux method. In 1-dimension, this is equivalent to factorisation of the Schrödinger operator, leading to the well known Infeld-Hull potentials. In higher dimensions, the Schrödinger operator involves the Laplacian, which, on general manifolds is replaced by the Laplace-Beltrami operator. Factorisation is no longer possible, but Darboux transformations are applicable, but must be compatible with the isometries of the underlying space. In [2], Darboux transformations are constructed for the Laplace-Beltrami operator of a 2-dimensional space of constant curvature. For first order Darboux transformations, the resulting potentials are closely related to the KdV equation. Eigenfunctions of the Laplace-Beltrami operator are constructed through the representation theory of the symmetry algebra. Whilst this symmetry is destroyed by the Darboux transformations, some shadow remains in the form of second order commuting operators and the eigenfunctions are built from those of the Laplace-Beltrami operator through the Darboux transformations.

Super-integrability is the property of having more than the n first integrals required by Liouville's theorem. The maximal number is 2n-1, although only n of them can be in involution. An analogous property can hold for quantum integrable systems, but in terms of commuting operators. In the latter case it is possible to use these operators to build eigenfunctions in a way which is analogous to the highest weight construction of Lie algebras (see [3]). Particular cases lead to the explicit construction of Krall-Sheffer polynomials, which are 2-dimensional generalisations of the familiar orthogonal polynomials of mathematical physics.

References

1. A.P. Fordy, Symmetries, ladder operators and quantum integrable systems, Glasgow Mathematical Journal, 47A , 65-75, 2005.
2. A.P. Fordy, Darboux Related Quantum Integrable Systems on a Constant Curvature Surface J.Geom. and Phys., 56 , 1709-27, 2006.
3. A.P. Fordy, Quantum Super-Integrable Systems as Exactly Solvable Models, SIGMA , 3 , 025, 10 pages, 2007.

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Last Updated: 5th May, 2007.