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 $I(\psi )=\int (\frac{DvD{v}^{\ast }}{2}(x)-W(x))a(x{)}^{2}dx,$ \smallskip \noindent where $\psi$ is described by $\psi =a{e}^{i\frac{u}{h}}$, $u=\frac{v+v^{\ast }}{2},$ $a=e^{\frac{v^{\ast }-v}{2\hslash }}$, $v,v^{\ast }$ are real functions, $\int a^2(x)dx=1$, and $D$ is derivative on $x\in {\mathrm{T}}^n$. It is natural to consider the constraint $\mathrm{div}(a^{2}Du)=0$, 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 ${}^{\prime }=\frac{d}{d\tau }$. We show that the expression for the second variation of a critical solution is given by \smallskip $\int a^{2}D[v^{\prime }]D[(v^{\ast }{)}^{\prime }]dx.$ \smallskip Introducing the constraint $\int a^{2}Dudx=V$, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation.