Thesis title: Spectral Theory of Non-self-adjoint Dirac Operators and Other Dispersive Models
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 structure satisfying some rigidity assumptions. Employing resolvent estimates for the Schrödinger operator well-established in the literature, we can obtain, among others, the counterpart of the notorious results by A. A. Abramov, A. Aslanyan, and E. B. Davies, and by R. Frank, for the Dirac operator. In the massless case, we obtain the spectrum stability of the perturbed Dirac operator for any of our special potentials. It is remarkable that we now achieve results without assuming restrictions on the norm size of the potential; however, we dearly pay on the rigidity structure of the potential.
Many other results are presented in this chapter, concerning both the eigenvalues enclosure in (un)bounded regions and the spectrum stability, depending on the rigidity assumptions for the potential and involving different kinds of norms.
In the fourth chapter, we consider the Cauchy problem for the wave equation with scale-invariant damping and mass terms, power non-linearity, and small initial data.
We proceed recollecting, at the best of our knowledge, the many results achieved during the decades on the widely studied damped wave equation, with and without mass, reorganizing and unifying them, other than proving new results in the massive case (for the purely damped case, we find an improvement in the lifespan estimates in 1-dimension).
The main tool we will use is a Kato-type lemma, whose mechanism is essentially based on an inductive argument.
Our analysis wants to stress in particular the competition between the "wave-like" and "heat-like" behaviors of the solutions, not only respect to the critical power, but also respect to the lifespan estimates, hence exploring the "heat versus wave" antagonism in the blow-up context.
Finally, in the fifth chapter, we consider the generalized Tricomi equation, or Gellerstedt equation, with nonlinearity of derivative-type and small initial data. Very recently this equation catalyzed a lot of attention and many papers appeared about it in a short time. We will study the blow-up of this equation furnishing the papabili critical exponent and lifespan estimates. Of course, to confirm that they are indeed the right ones, further consideration should be done demonstrating existence results. An attempt in this direction is done here proving a local existence result by using Fourier estimates for the Taniguchi-Tozaki multipliers. As a consequence, we show the optimality of the lifespan estimates at least in 1-dimension.
This time, the main strategy relies on the construction of a suitable test function and hence applying the test function method in order to reach our claimed results.