| Summary: | In the present thesis, we are going to collect results belonging to two lines of research: the first part of the work is devoted to the spectral theory for non-self-adjoint operators, whereas in the second part we consider nonlinear hyperbolic equations with time depending coefficients, and in particular their blow-up phenomena. Both of them have been deeply explored for decades and are still highly topical nowadays, being fascinating both for the mathematical and physical community. The bulk of the thesis is constituted by five chapters, all almost completely self-contained, mirroring the five independent papers listed at the end of the Introduction. In the first chapter, we generalize in higher dimension a celebrated result by J.-C. Cuenin, A. Laptev, and C. Tretter on the compact localization for the eigenvalues of the Dirac operator perturbed by a possibly non-Hermitian potential, hence in the non-self-adjoint setting. Indeed, assuming the potential small respect to suitable mixed norms, we prove that the eigenvalues lie in two disks of the complex plane in the massive case, whereas the point spectrum is empty in the massless case. At this aim, we combine the Birman-Schwinger principle, a technique hugely employed in recent times after the seminal work by R. Frank, with some new Agmond-Hörmander-type estimates for the resolvent of the Schrödinger operator and its first derivatives. In the second chapter, again we take advantage of the main engine of the Birman-Schwinger operator fueled this time with resolvent estimates already published in the literature, but which imply spectral results for the Dirac operator (and for the Klein-Gordon one) worthy of consideration. Here, various results on the eigenvalues localization in compact sets and on the stability of the spectrum (even in the massive case) are achieved under smallness conditions with respect to dyadic norms and pointwise assumptions on the weighted potential. In the third chapter, we consider some families of potentials with a peculiar matricial ...
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