Spectral analysis of a q-difference operator

For a number q bigger than 1, we consider a q-difference version of a second-order singular differential operator which depends on a real parameter. We give three exact parameter intervals in which the operator is semibounded from above, not semibounded, and semibounded from below, respectively. We also provide two exact parameter sets in which the operator is symmetric and self-adjoint, respectively. Our model exhibits a more complex behavior than in the classical continuous case but reduces to it when q approaches 1.