The heat and Schr̈odinger equations on conic and anticonic-type surfaces
The heat and Schr̈odinger equations on conic and anticonic-type surfaces
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圆锥型和锑型表面上的热方程和薛定谔方程
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发表时间：
2013
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通讯作者：
Dario Prandi
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作者：
U. Boscain;Dario Prandi
本文研究了二维流形上的热和自由量子粒子（由薛定谔方程描述）的演化，流形具有退化的黎曼度量ds^2 = dx ^2 +| X| ^{-2alpha}d heta^2$，其中$xin mathbb{R}$，$ hetainmathbb{T}$和参数$alphainmathbb{R}$。对于$alpha-1$，这个度量描述了锥状流形（对于$alpha=-1 $，它是一个平锥）。对于$alpha=0$，它是一个圆柱体。对于1$的值，它是一个类似Grushin的度量。我们证明了Laplace-Beltrami算子$Delta$本质上是自伴的当且仅当$alpha
otin（-3，1）$.在这种情况下，唯一的自伴扩张是弗里德里希扩张$Delta_F$，它不允许热量和量子粒子通过奇异集${x=0}$进行通信。对于α in（-3，-1]$，我们证明了对于Schr“odinger方程，只有α in（-3，-1]$上的平均 波函数的heta$可以穿过奇异集，而热方程的唯一马尔可夫扩张的解（实际上是$Delta_F$）不能。对于α in（-1，1），我们证明了存在一个正则自伴扩张Delta_B，称为桥接扩张，它是马尔可夫的，并且允许通过奇点的完全通信（包括热和量子粒子）.此外，我们还研究了随机完备性（即，证明了$Delta_F$和$Delta_B$在奇点处是随机完备的当且仅当$Delta_F$在奇点处是随机完备的，而$Delta_B$在奇点处总是随机完备的.
We study the evolution of the heat and of a free quantum particle (described by the Schr"odinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{-2alpha}d heta^2$, where $xin mathbb{R}$, $ hetainmathbb{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 $alpha
otin(-3,1)$. 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 Schr"odinger equation only the average on $ heta$ 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.