Stability of steady states for Hartree and Schrödinger equations for infinitely many particles (Q2136417)

The present article is devoted to the study of the Cauchy problem \[ i \partial_t X = -\Delta X + (w * \mathbb{E}(|X|^2)) X, \quad X(t=0)=X_0, \] where \(X: \Omega \times \mathbb{R}_t \times \mathbb{R}^3_x \rightarrow \mathbb{C}\) is a random field defined over a probability space \((\Omega, \mathcal{A}, \mathbb{P})\) with expectation \(\mathbb{E}\), \(*\) is the convolution product in \(\mathbb{R}^3\), and \(w\) is a finite Borel measure on \(\mathbb{R}^3\). This equation is closely related to the Hartree equation that is used to study systems of infinitely many Fermions. For any Wiener process \(W\) of dimension \(3\) such that \(\mathbb{E}(dW(\xi) \overline{dW(\eta)}) = \delta(\xi-\eta) d\xi d\eta\) and \(f \in L^2(\mathbb{R}^3, \mathbb{C})\) the random field \[ Y_f: \Omega \times \mathbb{R}_t \times \mathbb{R}_x^3 \rightarrow \mathbb{C}, \quad (\omega, t, x) \mapsto \int_{\mathbb{R}^3} f(\xi) e^{-it(m+\xi^2)+i \xi \cdot x} dW(\xi)(\omega), \] with \(m = \int_{\mathbb{R}^3} w \int_{\mathbb{R}^3} |f|^2\) is a steady state solution of the above equation. The main result of the paper deals with small perturbations of these steady states: under suitable assumptions on \(f\) and \(w\), it is shown that the above equation is well posed for initial conditions that are suitable perturbations of \(Y_f(t=0)\). Relating the framework of random fields to that of density matrices, also a scattering result for the operator \(\mathbb{E}(|X \rangle \langle X|)\) is obtained. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations. Eventually, similar results in dimension two are shown as well.