Exact integrability conditions for cotangent vector fields

Abstract

In Quantum Hydro-Dynamics the following problem is relevant: let $(\sqrt{\rho },\Lambda )\in H^1({\mathbb{R}}^d)\times L^2({\mathbb{R}}^d,{\mathbb{R}}^d)$ be a finite energy hydrodynamics state, i.e. $\Lambda =0$ when $\rho =0$ and \begin{equation*} E = \int_{\R^d} \frac{1}{2} \big| \nabla \sqrtρ \big|^2 + \frac{1}{2} Λ^2 \mathcal L^d < \infty. \end{equation*} The question is under which conditions there exists a wave function $ψ\in H^1(\R^d,\C)$ such that \begin{equation*} \sqrtρ = |ψ|, \quad J = \sqrtρ Λ= \Im \big( \bar ψ\nabla ψ). \end{equation*} The second equation gives for $\psi =\sqrt{\rho }w$ smooth, $|w|=1$, that $i\Lambda =\sqrt{\rho }\bar{w}\nabla w$. Interpreting $\rho {\mathcal{L}}^d$ as a measure in the metric space ${\mathbb{R}}^d$, this question can be stated in generality as follows: given metric measure space $(X,d,\mu )$ and a cotangent vector field $v\in L^2(T^{\ast }X)$, is there a function $w\in H^1(\mu ,{\mathbb{S}}^1)$ such that \begin{equation*} dw = i w v. \end{equation*} %dw = i w v$?Weshowthatundersomeassumptionsonthemetricmeasurespace$(X,d,μ)$(conditionswhichareverifiedonRiemannmanifoldswiththemeasure$μ= ρ\mathrm{Vol}$ormoregenerallyonnon-branching$MCP(K,N)$),weshowthatthenecessaryandsufficientconditionsfortheexistenceof$w$ is that (in the case of differentiable manifold) \begin{equation*} \int v(γ(t)) \cdot \dot γ(t) dt \in 2π\Z \end{equation*} for $π$-a.e.$γ$,where$π$isatestplansupportedonclosedcurves.Thisconditiongeneralizestheconditionsthatthevorticityisquantized.Wealsogivearepresentationofeverypossiblesolution.Inparticular,wededucethatthewavefunction$ψ= \sqrtρ w$isin$W^{1,2}(X)$whenever$\sqrtρ \in W^{1,2}(X)$.

32 pages, 1 figure