110 research outputs found
Quasi-exactly solvable problems and the dual (q-)Hahn polynomials
A second-order differential (q-difference) eigenvalue equation is constructed
whose solutions are generating functions of the dual (q-)Hahn polynomials. The
fact is noticed that these generating functions are reduced to the (little
q-)Jacobi polynomials, and implications of this for quasi-exactly solvable
problems are studied. A connection with the Azbel-Hofstadter problem is
indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed,
to appear in J.Math.Phy
An SU(2) Analog of the Azbel--Hofstadter Hamiltonian
Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and
Kashaev concerning the appearence of the quantum U_q(sl(2)) symmetry in the
problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce
a modification of the tight binding Azbel--Hofstadter Hamiltonian that is a
specific spin-S Euler top and can be considered as its ``classical'' analog.
The eigenvalue problem for the proposed model, in the coherent state
representation, is described by the S-gap Lam\'e equation and, thus, is
completely solvable. We observe a striking similarity between the shapes of the
spectra of the two models for various values of the spin S.Comment: 19 pages, LaTeX, 4 PostScript figures. Relation between Cartan and
Cartesian deformation of SU(2) and numerical results added. Final version as
will appear in J. Phys. A: Math. Ge
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