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 Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric ds2=dx2+|x|−2αdθ2ds2=dx2+|x|−2αdθ2, where x∈Rx∈R, θ∈Tθ∈T and the parameter α∈Rα∈R. For α≤−1α≤−1 this metric describes cone-like manifolds (for α=−1α=−1 it is a flat cone). For α=0α=0 it is a cylinder. For α≥1α≥1 it is a Grushin-like metric. We show that the Laplace–Beltrami operator Δ is essentially self-adjoint if and only if α∉(−3,1)α∉(−3,1). In this case the only self-adjoint extension is the Friedrichs extension ΔFΔF, that does not allow communication through the singular set {x=0}{x=0} both for the heat and for a quantum particle. For α∈(−3,−1]α∈(−3,−1] we show that for the Schrödinger equation only the average on θ   of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ΔFΔF) cannot. For α∈(−1,1)α∈(−1,1) we prove that there exists a canonical self-adjoint extension ΔBΔB, 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 L1L1 norm for the heat equation) of the Markovian extensions ΔFΔF and ΔBΔB, proving that ΔFΔF is stochastically complete at the singularity if and only if α≤−1α≤−1, while ΔBΔB is always stochastically complete at the singularity.