Linear response theory for Gross Pitaevskii equation

Question

I am trying to linearize the following GP eq: \begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\nabla^{2}}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r)\right]\psi(r,t) \end{equation}

The ansatz for the mean-field wavefunction is: \begin{equation} \psi_{0}(r,t)=\psi_{0}\, e^{i(k_{0}r-\omega_{0}t)} \end{equation}

One then has to add the fluctuations on top: \begin{equation} \psi(r,t)=\big[\psi_{0}(r)+\delta\psi(r,t)\big]\, e^{-i\omega_{0}t} \end{equation}

Pluggin this into the original equation we get to 0 order that \begin{equation} \omega_{0}-\frac{k_{0}^{2}}{2m}=g|\psi_0|^2. \end{equation}

Expanding to first order (linear response) we get \begin{equation} i\partial_{t}\delta\vec{\psi}=\mathcal{L}\cdot\delta\vec{\psi}+\vec{F}_{d} \end{equation}

with \begin{equation} \delta\vec{\psi}(r,t)=\left(\begin{array}{c} \delta\psi(r,t)\\ \delta\psi^{\star}(r,t) \end{array}\right) \end{equation}

\begin{equation} \vec{F}_{d}(r)=V_{d}(r)\,\left(\begin{array}{c} \psi_{0}(r)\\ -\psi_{0}^{*}(r) \end{array}\right) \end{equation}

\begin{equation} \mathcal{L}=\left(\begin{array}{cc} -\frac{k_{0}^{2}}{2m}-\frac{\nabla^{2}}{2m}+g|\psi_{0}|^2 & g\psi_{0}^{2}\, e^{2ik_{0}r}\\ -g\psi_{0}^{2\star}\, e^{-2ik_{0}r} & -\left(-\frac{k_{0}^{2}}{2m}-\frac{\nabla^{2}}{2m}+g|\psi_{0}|^2\right) \end{array}\right) \end{equation}

The goal here is to determine $\delta\vec{\psi}(r,t)$, by diagonalizing $\mathcal{L}$ and expanding on the corresponding eigenmodes. I am trying to follow these notes http://arxiv.org/abs/cond-mat/0105058v1, which give the general formalism starting on page 66.

However, the author says $\mathcal{L}$ is not diagonalizable in general therefore one has to do the trick of splitting $\delta\psi$ into a part along $\psi_0$ and a part orthogonal to it (see eq. 229). This leads to a new operator $\mathcal{L}$, given by (235) which is diagonalizable.

How can I apply this formalism to my problem? How do I construct the new operator $\mathcal{L}$? How do I deal with the projection operators (233) and (234), etc

Answer 1

Apologies for a very late response. I recently joined this community and am posting this answer for the benefit of the OP of this unanswered question.

The decay of a density of particles governed by stochastic differential equation (SDE) dynamics to the equilibrium density is a well known topic in statistical mechanics, for which you may see the book by Risken. For the stochastic process $x_s \sim p(s,x)$, for all $0 \leq s$, with the SDE dynamics $d x_s = b(x_s) ds + \sigma dw_s$ and the corresponding linear Fokker-Planck (FP) partial differential equation (PDE) dynamics $\partial_t p = \mathcal{L}^\dagger p$, the question is whether any initial density will decay to the invariant density $p^\infty(x)$ which solves the stationary FP equation and at what rate? This problem has been answered for certain systems, namely, Ornstein-Uhlenbeck and Langevin systems. The key idea is to consider the initial density as a perturbation from the equilibrium given as $p(0,x) = p^\infty(x) + p^\infty(x) \tilde{p}(0,x)$ and prove that $\tilde{p}(s,x)$ vanishes under the limit $s \rightarrow +\infty$, by using the eigen properties of the generator $\mathcal{L}$ of the SDE which is indeed the adjoint of the forward FP operator $\mathcal{L}$. Thus, properties of the SDE dynamics determine whether or not we can show that the equilibrium density is indeed attained by particles with any arbitrary initial density.

Similarly, in the case of the linear Schrödinger equation, eigen properties and corresponding stability results have been explored, in particular please consider the book by Berezen on this topic, with comprehensive exploration for various classes of Schrödinger potentials. More recently, this analysis has found application in the theory of mean field games (MFGs) starting with the seminal work in the integrator dynamics regime (similar to the Ornstein-Uhlenbeck case in the physics literature) and more recently in the case of Langevin dynamics. The former work considers the case of simple integrator SDE dynamics for a particular class of MFGs, which shows strong analogies with the non-linear Schrödinger and explicitly treating the Gross-Pitaevskii equation. The latter work is more general in terms of the nonlinear Langevin SDE dynamics for a general class of large-population systems, but shows stability results using the linear Schrödinger equation.

More specifically, the cited references indeed treat the diagonalization of PDE systems which correspond to that mentioned in the original question, with the key idea being the linearization and subsequent linear analysis using the eigen properties of the linear operators governing the local dynamics close to the equilibrium.