Exact eigenvalues of the Hamiltonian P²+A mod X mod ^nu

The correspondence between operators on a Hilbert space and phase-space functions based upon symmetric ordering, introduced by Cahill and Glauber (1969) (Weyl correspondence) is used in the present paper to define (non-linear) unitary transformations for quantum systems with the help of canonical transformations, which are bijective, i.e. one-to-one onto. These unitary transformations can be used to determine exactly the energy eigenvalues of a large class of one-dimensional quantum systems. As an example the authors calculate the exact eigenvalues of the Hamiltonian H(X,P)=¹/_2m(P²+a¹2( nu +2)/ mod X mod ^nu), a, nu >0.