An integral transform and its applications

Let p be the operator in L-2(0,pi): p=i(sin theta d/dtheta + 1/2 cos theta), 0less than or equal tothetaless than or equal topi. The operator p appears in quantum mechanics on the 1-sphere S-1 based on Dirac formalism. We show selfadjointness of the operator p and then construct an integral transform W satisfying WpW*=k, where W:L-2(0,pi)-->L-2(R) is a unitary operator and k denotes the multiplication by k (kis an element ofR), i.e., k:U(k)-->kU(k), Uis an element ofL(2)(R). We apply the transform to the Cauchy problems for some partial differential equations and explicitly find out the solutions. Moreover, we apply the transform to the study of quantum mechanics on the 1-sphere S-1 based on Dirac formalism.