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- W1576793966 abstract "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 $ds^2=dx^2+|x|^{-2alpha}dtheta^2$, where $xin mathbb{R}$, $thetainmathbb{T}$ and the parameter $alphainmathbb{R}$. For $alphale-1$ this metric describes cone-like manifolds (for $alpha=-1$ it is a flat cone). For $alpha=0$ it is a cylinder. For $alphage 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $Delta$ is essentially self-adjoint if and only if $alphanotin(-3,1)$. 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 $alphain(-3,-1]$ 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 $alphain(-1,1)$ 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 $alphale -1$, while $Delta_B$ is always stochastically complete at the singularity." @default.
- W1576793966 created "2016-06-24" @default.
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- W1576793966 date "2013-05-22" @default.
- W1576793966 modified "2023-09-27" @default.
- W1576793966 title "The heat and Schr̈odinger equations on conic and anticonic-type surfaces" @default.
- W1576793966 hasPublicationYear "2013" @default.
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