Lamé and ZK: spectral analysis and unique continuation

The bulk of this thesis focuses on two fields that are being deeply investigated both by the mathematical and physical community, namely spectral theory and unique continuation. Both these theories are extremely rich and nowadays represent inclusive terms covering a wide variety of branches of physics and mathematics. More precisely, the first one, in its general meaning, includes theories which extend the eigenvalues analysis for square matrices to a much broader class of mathematical characters, for instance, due to their relevance in quantum interpretation, to unbounded operators in Hilbert space. The second one is concerned with the search for classes of functions for which the vanishing in a region ensures the vanishing in a larger one, roughly speaking it is the issue to find the correct analogue of harmonic functions for which the Liouville theorem guarantees the stated rigidity. The structure of this document is roughly the following. The main body of work of this thesis is contained in the first two parts, in which the aforementioned themes are analyzed, specifically spectral properties for the non self-adjoint perturbed Lamé operator of elasticity and unique continuation for Zakharov-Kuznetzov dispersive equation are objects of our investigation. Each part contains an introductory chapter which endeavors to give an overview of the problem in exam and to clarify why it is worthy of attention. Moreover a time-based analysis, involving also the recent developments of these matters, is provided in the same chapters. At times the discussion is chosen to be informal in order to convey the basic underlying ideas. The concise statements together with their proofs, employing the necessary rigor lacking in the introductions, are given in the following chapters. The third part is slightly different, it is not concerned with achieved results but it involves a future possible project that we would like to deepen. More precisely the prospect presented takes place in the field of inverse problems in elasticity. The possibility to re-adapt some useful tools earned to address the problems described in the first two parts has played a relevant role to motivate solidly this future investigation.