On the quantum Guerra-Morato action functional

Given a smooth potential W : T n → R on the torus, the Quantum Guerra-Morato action functional is given by I ( ψ ) = ∫ ( D v D v * 2 ( x ) − W ( x ) ) a ( x ) 2 d x , where ψ is described by ψ = a e i 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.