An extended q - deformed su ( 2 ) algebra and the Bloch electron problem

It is shown that an extended q-deformed su(2) algebra with an extra (“Schwinger ”) term can describe Bloch electrons in a uniform magnetic field with an additional periodic potential. This is a generalization of the analysis of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory of this q-deformed algebra, we obtain functional Bethe ansatz equations whose solutions should be functions of finite degree. It is also shown that the zero energy solution is expressed in terms of an Askey Wilson polynomial. The quantum deformation [1] of algebras was introduced in connection with inverse scattering problems and integrable models. It is also known that this notion has a deep relation to Yang-Baxter equations [2]. The Uq(sl2) algebra, one of the q-deformed algebras, was applied to an analysis of the two-dimensional Bloch electron problem [3] by Wiegmann and Zabrodin [4]. They found functional Bethe ansatz equations for this problem. From the representation theory of Uq(sl2) algebra on the functional space, they deduce that the solutions of the Bethe ansatz equations should be functions of finite degree. The zero energy solutions and the generalized functional Bethe ansatz equations have also been discussed in Refs. [5][6]. Biedenharn [7] and Macfarlane [8] constructed Uq(sl2) algebra, in the manner of Schwinger’s construction of conventional su(2), by using two sets of qdeformed oscillator algebras. The q-deformed oscillator algebra Hq(1), if suitably defined, ∗ E-mail address : [email protected]