2013/05/31 by Ugo Boscain, Dario Prandi
Economics, Econometrics and Finance · Mathematics · #Conic section #Degenerate energy levels #Extension (predicate logic) #Geometry #Gravitational singularity #Heat equation #Hilbert space #Laplace operator #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum mechanics #Self-adjoint operator #Singularity #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #advanced mathematical theories #math.AP
paper · pdf · doi:10.1016/j.jde.2015.10.011
published as Journal of Differential Equations. Volume 260, Issue 4, 15 February 2016, Pages 3234-3269 · 29 pages, 2 figures, accepted version
arxiv created 2015/10/07 · openalex publication_date 2015/11/10 · openalex created_date 2016/06/24 · arxiv updated 2019/06/21 · openalex updated_date 2026/08/05
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|-2αdθ2, where x∈ \mathbb R, θ∈\mathbb T and the parameter α∈\mathbb R. For α≤-1 this metric describes cone-like manifolds (for α=-1 it is a flat cone). For α=0 it is a cylinder. For α≥ 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). In this case the only self-adjoint extension is the Friedrichs extension ΔF, that does not allow communication through the singular set \x=0\ both for the heat and for a quantum particle. For 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 ΔF) cannot. For α∈(-1,1) we prove that there exists a canonical self-adjoint extension ΔB, 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 L1 norm for the heat equation) of the Markovian extensions ΔF and ΔB, proving that ΔF is stochastically complete at the singularity if and only if α≤ -1, while ΔB is always stochastically complete at the singularity.