A Polynomial-Type Jost Solution and Spectral Properties of a Self-Adjoint Quantum-Difference Operator

Abstract In this paper, we find a polynomial-type Jost solution of a self-adjoint %$q%$-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator %$L%$ generated by the %$q%$-difference exp...
Ausführliche Beschreibung

Abstract In this paper, we find a polynomial-type Jost solution of a self-adjoint %$q%$-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator %$L%$ generated by the %$q%$-difference expression of second order has essential spectrum filling the segment %$[-2\sqrt{q},2\sqrt{q}]%$, %$q>1%$. Finally, we examine the properties of the eigenvalues of %$L%$. Ausführliche Beschreibung