A singular Lambert-W Schr\"odinger potential exactly solvable in terms of the confluent hypergeometric functions

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schr\"odinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schr\"odinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one it supports only a finite number of bound states

Constant mean curvature surfaces and Heun's differential equations

This thesis is concerned with the problem of constructing surfaces of constant mean curvature with irregular ends by using the class of Heun’s Differential Equations. More specifically, we are interested in obtaining immersion of punctured Riemann spheres into three dimensional Euclidean space with constant mean curvature. These immersions can be described by a Weierstrass representation in terms of holomorphic loop Lie algebra valued 1-forms. We describe how to encode each of the differential equations in Heun’s family in the Weierstrass representation. Next, we investigate monodromy problems for each of the cases in order to ensure periodicity of all the resulting immersions. This allows us to find four families of surfaces with constant mean curvature and irregular ends. These families can be described as trinoids, cylinders, perturbed Delaunay surfaces and planes. Finally, we study some symmetry properties of these groups of surfaces

Subtracted Geometry

In this thesis we study a special class of black hole geometries called subtracted geometries. Subtracted geometry black holes are obtained when one omits certain terms from the warp factor of the metric of general charged rotating black holes. The omission of these terms allows one to write the wave equation of the black hole in a completely separable way and one can explicitly see that the wave equation of a massless scalar field in this slightly altered background of a general multi-charged rotating black hole acquires an $SL(2,\mathbb{R}) \times \times SL(2,\mathbb{R}) \times SO(3)$ symmetry. The ”subtracted limit” is considered an appropriate limit for studying the internal structure of the non-subtracted black holes because new \u27subtracted\u27 black holes have the same horizon area and periodicity of the angular and time coordinates in the near horizon regions as the original black hole geometry it was constructed from. The new geometry is asymptotically conical and is physically similar to that of a black hole in an asymptotically confining box. We use the different nice properties of these geometries to understand various classically and quantum mechanically important features of general charged rotating black holes

Time-dependent Quantum Systems

This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.Siirretty Doriast

Lectures on Applied Mathematics Part 2: Numerical Analysis

This book is designed to be a continuation of the textbook, Lectures on Applied Mathematics Part I: Linear Algebra which can also be downloaded at http://rbowen.engr.tamu.edu. This textbook evolved from my teaching an undergraduate Numerical Analysis course to Mechanical Engineering students at Texas A&M University. That course was one of the courses I was allowed to teach after my several years out of the classroom. It tries to utilize rigorous concepts in Linear Algebra in combination with the powerful computational tools of MATLAB to provide undergraduate students practical numerical analysis tools. It makes extensive use of MATLAB's graphics capabilities and, to a limited extent, its ability to animate the solutions of ordinary differential equations. It is not a textbook that tries to be comprehensive as a source of MATLAB information. It does contain a large number of links to MATLAB's extensive online resources. This information has been invaluable to me as this work was developed. The version of MATLAB used in the preparation of this textbook is MATLAB 2019b.Chapter 7: Elements of Numerical Linear Algebra; Chapter 8: Errors that Arise in Numerical Analysis; Chapter 9: Roots of Nonlinear Equations; Chapter 10: Regression; Chapter 11: Interpolation; Chapter 12: Ordinary Differential Equations; Appendix A: Introduction to MATLAB; Appendix B: Animation