Year: 2024
Type: article
Abstract: Given a smooth potential W:Tn→R on the torus, the Quantum Guerra–Morato action functional is given by I(ψ)=∫(DvDv*2(x)−W(x))a(x)2dx, where ψ is described by ψ=aeiuℏ, u=v+v*2, a=ev*−v2ℏ, 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 ′ =ddτ. We show that the expression for the second variation of a critical solution is given by ∫a2D[v′] D[(v*)′] dx. Introducing the constraint ∫a2Du dx = V, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation.
Source: Journal of Mathematical Physics
DOI: 10.1063/5.0207422
Authors: Josué Knorst, Artur O. Lopes
Institution: Universidade Federal do Rio Grande do Sul
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4401723214