On the weakly disordered Anderson model and aspects of its kinetic scaling limit to the linear Boltzmann equation

We consider a random Schrödinger equation describing a quantum mechanical particle under a weak Gaussian random potential in ℝ [superscript d] with d ≥ 3. We prove a partial result on the conjectural convergence to a linear Boltzmann equation of the expectation of its Wigner transform in macroscopic spacetime coordinates. Our conjecture is closely related to the result obtained by Erdõs and Yau, [Comm. Pure Appl. Math. 53 (6), (2000)]. Whereas the result by Erdõs and Yau proves the convergence of the expected Wigner transform to a solution to a linear Boltzmann equation for any fixed time, our conjecture is that the time evolution of the expected Wigner transform is governed in the limit by the linear Boltzmann equation. We show that the conjecture is equivalent to the vanishing in the limit of the error incurred when keeping certain terms under a formal Wick ordering argument, and propose a method to tackle this equivalent statement. We also provide rigorous interpretations of the expectation and its time derivative of a random Wigner transform.