Exact integrability conditions for cotangent vector fields

TL;DR

This paper establishes precise integrability conditions for cotangent vector fields to be represented as phase gradients of wave functions in quantum hydrodynamics, generalizing quantization conditions on metric measure spaces.

Contribution

It provides necessary and sufficient conditions for the existence of wave functions with prescribed cotangent fields on metric measure spaces, extending previous results to non-smooth settings.

Findings

Conditions for wave function existence involve integral quantization over closed curves.

The results apply to Riemannian manifolds and non-branching MCP(K,N) spaces.

Every solution can be explicitly represented, ensuring wave functions are in W^{1,2} when density is in W^{1,2}.

Abstract

In Quantum Hydro-Dynamics the following problem is relevant: let $(\sqrt{\rho},\Lambda) \in H^1(\R^d) \times L^2(\R^d,\R^d)$ 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 \sqrt{\rho} \big|^2 + \frac{1}{2} \Lambda^2 \mathcal L^d < \infty. \end{equation*} The question is under which conditions there exists a wave function $\psi \in H^1(\R^d,\C)$ such that \begin{equation*} \sqrt{\rho} = |\psi|, \quad J = \sqrt{\rho} \Lambda = \Im \big( \bar \psi \nabla \psi). \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 $\R^d$, this question can be stated in generality as follows: given metric measure space $(X,d,\mu)$ and a cotangent vector field $v \in L^2(T^* X)$, 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$? We show that under some assumptions on the metric measure space $(X,d,\mu)$ (conditions which are verified on Riemann manifolds with the measure $\mu = \rho \mathrm{Vol}$ or more generally on non-branching $MCP(K,N)$), we show that the necessary and sufficient conditions for the existence of $w$ is that (in the case of differentiable manifold) \begin{equation*} \int v(\gamma(t)) \cdot \dot \gamma (t) dt \in 2\pi \Z \end{equation*} for $\pi$-a.e. $\gamma$, where $\pi$ is a test plan supported on closed curves. This condition generalizes the conditions that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function $\psi = \sqrt{\rho} w$ is in $W^{1,2}(X)$ whenever $\sqrt{\rho} \in W^{1,2}(X)$.