A Theoretical Investigation of Time-Dependent Kohn--Sham Equations

Abstract

In this work, the existence, uniqueness and regularity of solutions to the time-dependent Kohn-Sham equations are investigated. The Kohn-Sham equations are a system of nonlinear coupled Schrödinger equations that describe multi-particle quantum systems in the framework of the time dependent density functional theory. In view of applications with control problems, the presence of a control function and of an inhomogeneity are also taken into account. * Supported in part by the Deutsche Forschungsgemeinschaft (DFG) project "Controllability and Optimal Control of Interacting Quantum Dynamical Systems" (COCIQS).

A theoretical investigation of time-dependent Kohn–Sham equations: new proofs

“…with δΨ(0) = 0. The energy estimate (16) remains valid for (25) and implies that δΨ Y = 0, which is a contradiction. Hence, the uniqueness of Ψ follows.…”

“…and V xc (Ψ(x, t)) ∈ R takes into account exchange and correlation potentials, whose dependence on time and space variables is implicit through Ψ. For further details about the Kohn-Sham model, we refer to [25] and references therein. In the Hartree potential, we introduced the operator E that extends Ψ from Ω to R 3 .…”

“…To the best of our knowledge, a time-dependent model similar to the one considered in this work, is only addressed in [14,15,16,25]. In [14,15,16] the authors prove existence and uniqueness of solutions assuming that the potentials are continuously differentiable in time.…”

“…As mentioned above, this assumption appears too strong in the context of control applications. An attempt to improve these results is made in [25], where the authors try to obtain existence and regularity results by exploiting a Galerkin approach combined with energy estimates. Unfortunately, the proofs of some energy estimates derived in [25] are erroneous.…”

“…Finally, we wish to remark that the results presented in this work can be extended to the case of the Kohn-Sham adjoint equation, which is used in the framework of optimal control problems governed by the TDKS equation. This adjoint equation can be regarded, in some sense, as a linearized TDKS equation with inhomogeneous righthand side; see, e.g., [24,26,25].…”