Spectral and dynamical properties of quantum systems

Abstract. The leitmotif of this work is the interplay between the spectral properties of a closed quantum system and its evolution. The essay is divided into two thematically distinct, but strictly intertwined, parts.

Part I, General topics in quantum dynamics, collects some results about the evolution of the survival amplitudes of states of a quantum system, as well as the expectation values of quantum observables. Chapter 1 provides a warm-up discussion about the central topic of this essay, listing some classic results. Chapter 2 examines in detail a strikingly general feature of quantum systems, that is, large-time deviations from the exponential decay, generalizing a known argument by Khalfin. In Chapter 3 we finally embrace the description of quantum dynamics via the Fourier-Laplace transform, using the tools of complex analysis, to show that, under general assumptions, the survival amplitude of a quantum state can be decomposed into three kinds of contributions with different dynamical behavior. The same tools are used in Chapter 4 to show a somewhat surprising feature of quantum evolution laws, that is, their finite-time indistinguishability; an interesting application of the latter phenomenon is discussed.

Part II, Friedrichs-Lee Hamiltonians and applications, is centered on an interesting class of exactly solvable models, which we denote as Friedrichs-Lee Hamiltonians. In Chapter 5 we introduce such models, generalize their structure in such a way to accommodate, in a fully rigorous way, a far larger class of physically interesting models, also leading us to a critical discussion about a seemingly unrelated concept borrowed from quantum field theory: renormalization. This analysis is completed in Chapter 6, which is devoted to the analysis of their spectral properties. In the last chapters we analyze some particular Friedrichs-Lee models describing the interaction of quantum emitters with a properly structured one-dimensional boson bath, chosen in such a way to reproduce an electromagnetic field confined in a waveguide: Chapter 7 analyzes a single quantum emitter in a infinite or semi-infinite waveguide, while Chapter 8 focuses on the insurgence of many-body bound states in the continuum (BICs) for an array of quantum emitters in an infinite waveguide.