Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces

We study the evolution of the heat and of a free quantum particle (described by the Schrodinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric ds(2) = dx(2) + vertical bar x vertical bar(-2 alpha)d theta(2), where x is an element of R, theta is an element of T and the parameter alpha is an element of R. For alpha <=-1 this metric describes cone-like manifolds (for alpha = -1 it is a flat cone). For alpha = 0 it is a cylinder. For alpha >= 1 it is a Grushin-like metric. We show that the Laplace-Beltrami operator Delta is essentially self-adjoint if and only if alpha is not an element of (-3, 1). In this case the only self-adjoint extension is the Friedrichs extension Delta(F), that does not allow communication through the singular set {x = 0} both for the heat and for a quantum particle. For alpha is an element of(-3, 1] we show that for the Schrodinger equation only the average on theta of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is Delta(F)) cannot. For alpha is an element of (-1, 1) we prove that there exists a canonical self-adjoint extension Delta(B), called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the L-1 norm for the heat equation) of the Markovian extensions Delta(F) and Delta(B), proving that Delta(F) is stochastically complete at the singularity if and only if alpha <= -1, while Delta(B) is always stochastically complete at the singularity. (C) 2015 Elsevier Inc. All rights reserved.