Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators

We consider a one-dimensional generalized diffusion operator G on an open interval which is not necessarily bounded. We characterize [SP] (i.e., 1 is a small perturbation of the operator G) in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for [IU] (i.e., the associated heat kernel is intrinsically ultracontractive).