Applications of a novel approach to relativistic and non-relativistic problems in physics

We have developed an algebraic approach for the treatment of time-independent Schrödinger equation with constant/non-constant masses within the frame of non-relativistic quantum theory. The model developed then has been successfully applied in various fields of physics involving exactly/approximately solvable potentials. The first part of the thesis work is devoted to the presentation of these applications, such as the application results for noncentral potentials, applicability of the scheme for scattering theory and careful analysis of the application results for quantum systems with position-dependent masses in arbitrary dimensions After gaining confidence from the successful applications of this novel formalism to non-relativistic systems, we have extended our investigations by studying the applicability of the model also for the relativistic considerations in the light of Klein-Gordon and Dirac equations involving only bound quantum states. These considerations require naturally the split of the relativistic equations into two parts, unlike the other models in the literature, which provide a clear visualization of the relativistic contributions, in an explicit manner, to the solution in the non-relativistic limit. Key words: Schrödinger Equation, Dirac Equation, Klein-Gordon Equation, Exactly-solvable Systems, Non-central Potentials, Position-dependent Mass.