Quantum kinetic models of open quantumsystems in semiconductors theory

From the introduction: The aim of the developed research is to contribute to the analytical study of quantum kinetic models of certain quantum systems, whose dynamics is time-irreversible due to the interaction with the environment; accordingly, they are called open. In particular, the models under examination have a well-grounded application to the simulation of nanoscale semiconductor devices, thus semiconductor physics is the background of reference for our work. The models are formulated according to the Wigner-function formalism, a well-known tool in both the physical and the mathematical literatures, which provides a quantum-mechanically consistent, phase-space description of the dynamics of the systems of interest. The leitmotiv of our investigation is the attempt of keeping to a purely kinetic analysis. More precisely, our aim is to obtain results that are physically-consistent and in agreement with those achieved via other formalisms, but independently of them. The way we pursue that end is by developing new analytical tools, in some cases inspired by formal analogies with other problems. The motivation for that type of study is that, apart from being analytically challenging, it is naturally suitable to numerical reformulation for applications to real devices simulation. The aspects of our research we have here presented will be widely discussed, in comparison with the existing literature, according to the following sectioning: In Part I, we will present an overview of the possible mathematical descriptions of semiconductor physics. In particular, in Chapter 1, we will introduce the (semi-)classical kinetic approach and start to discuss possible ways of modelling the irreversible dynamics of certain systems. In Chapter 2, we will focus on quantum systems, since we are interested in the applications to semiconductor devices reaching quantum regimes and we will present the quantum-statistical formalism. In Section 2.2 we will briefly introduce the theory of open quantum systems, which constitutes the reference frame of our work, and proceed in the discussion of the literature related to the description of irreversible quantum dynamics. Thus, the background is complete. The aim of Chapter 3, in Part II, is to present the quantum kinetic description of the quantum systems, and, to compare it with the quantum-statistical one. By that discussion we will derive the motivation for our analytical study: in particular, in Section 3.4, we will describe the starting point of our investigation, namely, the choice of the Hilbert space of L2-functions defined on the phase-space, as the state space for the successive well-posedness studies. In Chapter 4 are introduced new tools we have employed in the cited articles, which are also promising in view of the resolution of open problems. We will compare them with similar instruments used in classical kinetic theory, which in many cases have directly inspired their derivation in the quantum framework. The tools presented constitute a contribution to the discussion in literature about the analogies between the Schr¨odinger and the Vlasov equations. We anticipate that we will recover a further a posteriori motivation for our choice of the functional setting: according to our investigation, the analogy with the classical kinetic formalism can indeed be exploited just in the L2-context. Part III and Part IV contain the bodies of the above cited articles: in particular, Part III those related to the Wigner-Poisson system on bounded spatial domains, while Part IV, the one about the (all-space) Wigner-Poisson-Fokker-Planck model. At the beginning of each part we will provide both a description of the physical system they are meant to describe and a discussion of the related literature. We remark that, in the three cases, the well-posedness result will be obtained in the Hilbert space of L2-functions defined on the phase-space, modified by an appropriate weight in the velocity-direction. Accordingly, the result we present in Part IV is a slightly improved version of the one presented in the above cited paper, where also a weight in the space-direction was used. Relatively to both problems (in the all-space formulation), we also discuss possible perspectives for attaining the same well-posedness theorems in a L2-setting, without using the weights.