标题： 关于量子Guerra-Morato作用泛函 显示英文标题

标题： On the quantum Guerra-Morato Action Functional

摘要： 给定一个在环面上的光滑势 $W:\mathrm{T}^{n} \to \mathbb{R}$，量子Guerra-Morato作用泛函由 \smallskip $ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, I(\psi) = \int\,(\, \, \,\frac{D v\, D v^*}{2}(x) - W(x) \,) \,\,a(x)^2 dx,$ \smallskip \noindent 其中 $\psi $ 由 $\psi = a\, e^{i\,\frac{ u }{h}} $描述， $ u =\, \frac{v + v^*}{2},$ $a=e^{\,\frac{v^*\,-\,v}{2\, \hbar} }$ ， $v,v ^*$是实函数， $\int a^2 (x) d x =1$， $D$是关于 $x \in \mathrm{T}^{n}$的导数。 考虑约束$ \mathrm{d}\mathrm{i}\mathrm{v}(a^{2}Du)=0$是自然的，这意味着通量为零。 从临界解（在变分$\tau$下）得到的$a$和$u$，满足带有量子势的哈密顿-雅可比方程。 记$'=\frac{d}{d\tau}$。 我们证明了临界解的二阶变分表达式为 \smallskip $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int a^{2}\,D[ v' ]\, D [(v ^*)']\, dx.$ \smallskip 引入约束 $\int a^2 \,D u \,dx =V$，我们之后还将考虑一个相关的对偶特征值问题。 由此得出一个传输方程和一种类似射线方程的方程。 显示英文摘要

摘要： Given a smooth potential $W:\mathrm{T}^{n} \to \mathbb{R}$ on the torus, the Quantum Guerra-Morato action functional is given by \smallskip $ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, I(\psi) = \int\,(\, \, \,\frac{D v\, D v^*}{2}(x) - W(x) \,) \,\,a(x)^2 dx,$ \smallskip \noindent where $\psi $ is described by $\psi = a\, e^{i\,\frac{ u }{h}} $, $ u =\, \frac{v + v^*}{2},$ $a=e^{\,\frac{v^*\,-\,v}{2\, \hbar} }$, $v,v ^*$ are real functions, $\int a^2 (x) d x =1$, and $D$ is derivative on $x \in \mathrm{T}^{n}$. It is natural to consider the constraint $ \mathrm{d}\mathrm{i}\mathrm{v}(a^{2}Du)=0$, which means flux zero. The $a$ and $u$ obtained from a critical solution (under variations $\tau$) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote $'=\frac{d}{d\tau}$. We show that the expression for the second variation of a critical solution is given by \smallskip $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int a^{2}\,D[ v' ]\, D [(v ^*)']\, dx.$ \smallskip Introducing the constraint $\int a^2 \,D u \,dx =V$, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation.