Non-Hermiticity in Quantum Physics

The field of non-Hermitian Physics has attracted great attention over the last 23years, both from the in the physical and mathematical communities. From the physical point of view, non-Hermiticity was regarded as a phenomenological tool to describe open quantum systems. Besides this, the rising interest in this field comes especially from the possible exploitation of exceptional points for quantum technologies, and from the exotic topology arising in periodic non-Hermitian systems, connected to the so called non-Hermitian skin effect. From the mathematical point of view, the range of possible topics to investigate has been wide open, as dropping an hypothesis of a theory makes the mathematician wonder what keeps holding true and what is lost. This dissertation wishes to be a contribution to both lines of research. In the first part, after a minimal introduction, we mainly discuss exotic behaviors at the exceptional point of two very different but realistic quantum systems: a gain-loss open system and a waveguide QED setup. The second part is devoted to a more mathematical approach to non self-adjoint operators inpired by quantum mechanics: the problem of quantization of a dissipative system is considered, and the construction of a class of non self-adjoint Hamiltonians is developed.