The modified scattering conjecture for critical and long-range density-functional interactions

Let $d$ be the spatial dimension, let $v(x)=|x|^{-\alpha}$ with $0<\alpha\leq 1$, and consider

$$g(\rho)=\lambda_1|x|^{-\alpha}\ast\rho+\lambda_2\rho^\beta,$$

with $0<\beta\leq 1/d$. Let $I^1$ denote the trace-class space used for the density matrices, and let $\gamma(t)$ be the solution. Modified scattering conjecture. Modified scattering holds. In particular, when $\alpha=1$ and $\beta=1/d$, there should exist a time-independent $\gamma_\infty\in I^1$ and a time-independent operator $g_\infty$ depending nonlinearly on $\gamma_\infty$ such that

$$\left\|\gamma(t)-e^{-i(-t\Delta+g_\infty(-i\nabla)\log t)}\gamma_\infty e^{i(-t\Delta+g_\infty(-i\nabla)\log t)}\right\|_{I^1}\longrightarrow0$$

as $t\to\infty$, with some algebraic rate. This conjecture concerns the long-range or critical regime not covered by the short-range scattering result, and specifies the expected nonlinear logarithmic phase correction.