On the quantum Guerra-Morato Action Functional

Abstract

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(ψ) = \int\,(\, \, \,\frac{D v\, D v^*}{2}(x) - W(x) \,) \,\,a(x)^2 dx,$ \smallskip \noindent where $ψ$ is described by $ψ= 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 $τ$) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote $'=\frac{d}{dτ}$. 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.

Theorems & Definitions (21)

• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 5
• proof
• Theorem 6
• Theorem 7
• proof
• Theorem 8