Self-adjointness of the momentum operator with a singular term

Self-adjointness is shown of the momentum operator p -i(d/dx) + i(c/x)R , Ru(x) = u(-x) , with domain {ul E I1I(Rl): ul/x E L2(R')} when c > 1 or c < . This operator appears in a harmonic oscillator system with the generalized commutation relations by Wigner: ip = [x, II] and -i-v = [p, II] for the Hamiltonian II and the multiplication operator x . The proof is carried out by generation of a unitary group in terms of ip, based on the Hille-Yosida theorem and Stone's theorem. -The result is applied to the self-adjoitness of II = (p2 + x2)/2.