On the quantum Guerra–Morato action functional

Abstract

Given a smooth potential W : Tⁿ → R on the torus, the Quantum Guerra–Morato action functional is given by I (ψ) = ∫ (Dv Dv * 2 (x) − W (x) ) a (x)² dx, where ψ is described by ψ = a ei u ℏ , u = v + v * 2 , a = e v * − v 2 ℏ , v, v* are real functions, ∫a2(x)dx = 1, and D is the derivative on x ∈ Tn. It is natural to consider the constraint div(a²Du) = 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 ′ = d d τ . We show that the expression for the second variation of a critical solution is given by ∫a² D[v′] D[(v*)′] dx. Introducing the constraint ∫a² Du dx = V, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation.