Time Dependent Closed Quantum Systems: Nonlinear Kohn-Sham Potential Operators and Weak Solutions

By: Joseph W. Jerome

We discuss time dependent quantum systems on bounded domains from the perspective of nonlinear, time-dependent potentials. The time dependence of the Kohn-Sham potentials distinguishes this study from that of the so-called nonlinear Schr\"{o}dinger equation, much studied in the mathematical community. We are interested in establishing a framework for potentials including the external potential, the Hartree potential and the exchange correlation potential as occur in time dependent density functional theory (TDDFT). As in previous work, we make use of the time-ordered evolution operator. A departure from the previous work is the use of weak solutions for the nonlinear model; this necessitates a new framework for the evolution operator based upon dual spaces. We are able to obtain unique global solutions. The author thanks Eric Polizzi for discussions leading to the incorporation of a version of the exchange correlation potential in the model.
This paper has been published in the Journal of Mathematical Analysis and Applications: vol. 429 (2015), pp. 995--1006. This preprint clarifies definitions in the journal article; we have removed the complex conjugation on the test functions involved in the definition of A(t) in section 2.1 and in the definition of S in section 2.2. Also, we have added a brief appendix to correct a `typo' in the smoothing hypothesis on the exchange-correlation potential. It can be viewed in the following format: