Specific non-relativistic quantum mechanical one-dimensional systems, interacting via point interactions, are discussed within the resolvent algebra setting.

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

The main bulk of this Thesis is focused on the response of cell membranes due to chemical and mechanical stimuli. Henceforth, it is mainly devoted to deduce how the key aspect of the cell response activated by chemical signaling can be predicted by a simplified energetics, making use of both theoretical models and numerical simulations. The a ention is focused on cell membranes embedding G protein-coupled receptors (GPRCs). By analyzing the behavior of cell mem- branes, one can isolate three main contributions in order to model their respon- se: (1) diffusion of receptors and transporters embedded in the lipid membrane; (2) conformational changes of the receptors; (3) membrane elasticity. Moreover, the interplay between TM confomational changes and lateral pressure of the lipid membrane against such TMs is introduced. The chemical potential of the receptor-ligand compound, deduced as the variational derivative of such energy, is compared with the one calculated by accounting for the work done by the lateral pressure. The result yields a relationship between the conformational field, the mechanical field (interpreted as either the thickness change or the areal change) and the distribution of the compounds receptor-ligand. The analysis of such resulting constitutive equation among those three quantities shows that, essentially, the reason why ligand-GPRCs compounds prefer to live on lipid ra is a necessity involving the interplay between the work performed by the lateral pressure and the need of TMs to change their conformation during ligand binding. Henceforth, mechanobiology gives a justification to the experimental findings of Kobilka and Lei ovitz, Chemistry Nobel Prizes 2012.