Search

In this thesis, we develop WKB techniques for the finite difference Schrodinger equation, following the construction of the WKB approach for the standard differential Schrodinger equation. In particular, we will develop an all-order WKB algorithm to get arbitrary hbar-corrections and construct a general quantum momentum, underlining the various properties of its coefficients and the quantities that will be used when constructing the quantization condition. In doing so, we discover the existence of additional periodic factors that need to be considered in order to obtain the most general solution to the problem at hand. We will then proceed to study the simplest non trivial example, the linear potential case and the Bessel functions, that provide a solution to the linear problem. After studying the resurgence properties of the Bessel functions from an analytical and numerical point of view, we will then proceed to use those results in order to build general connection formulae, allowing us to connect the local solutions defined on two sides of a turning point into a smooth solution on the whole real line. With those connection formulae, we will analyse a selection of problems, constructing the discrete spectrum of various finite difference Schrodinger problems and comparing our results with existing literature.

Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painleve I and Painleve II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work computes for the first time the complete, analytical, resurgent Stokes data for the first two Painleve equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed "closed-form asymptotics", makes sole use of resurgent large-order asymptotics of transseries solutions -- alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks -- hence for an analytical proof of our results., Comment: 122 pages, 51 plots in 40 figures, 19 tables, jheppub-nosort.sty; v2: small changes (additions, rewrites, typos); v3: more typos/corrections + more (iso)monodromy details in 7.3, 7.4

We discuss nuclear physics in the Witten-Sakai-Sugimoto model, in the limit of large number $N_c$ of colors and large 't Hooft coupling, with the addition of a finite mass for the quarks. Individual baryons are described by classical solitons whose size is much smaller than the typical distance in nuclear bound states, thus we can use the linear approximation to compute the interaction potential and provide a natural description for lightly bound states. We find the classical geometry of nuclear bound states for baryon numbers up to B=8. The effect of the finite pion mass - induced by the quark mass via the GMOR relation - is to decrease the binding energy of the nuclei with respect to the massless case. We discuss the finite density case with a particular choice of a cubic lattice, for which we find the critical chemical potential, at which the hadronic phase transition occurs., Comment: LaTeX: 35 pages, 5 figures; V2: comments and references added, typos corrected

We discuss nuclear physics in the Sakai-Sugimoto model in the limit of large number $N_c$ of colors and large 't Hooft coupling $\lambda$. In this limit the individual baryons are described by classical solitons whose size is much smaller than the typical distance at which they settle in a nuclear bound state. We can thus use the linear approximation outside the instanton cores to compute the interaction potential. We find the classical geometry of nuclear bound states for baryon number up to eight. One of the interesting features that we find is that holographic nuclear physics provides a natural description for lightly bound states when $\lambda$ is large. For the case of two nuclei, we also find the topology and metric of the manifold of zero modes and, quantizing it, we find that the ground state can be identified with the deuteron state. We discuss the relations with other methods in the literature used to study Skyrmions and holographic nuclear physics. We discuss $1/{N_c}$ and $1/\lambda$ corrections and the challenges to overcome to reach the phenomenological values to fit with real QCD., Comment: 47 pages, 7 figures. v2: typos corrected