Numerical and algebraic treatment of dynamical systems equation and soluable potentials

ABSTRACT NUMERICAL AND ALGEBRAIC TREATMENT OF DYNAMICAL SYSTEMS EQUATION AND SOLVABLE POTENTIALS EserKÖRCÜK M.Sc, Engineering Physics Department Supervisor: Assists. Prof. Dr. Ramazan KOÇ January, 2001, In this thesis, the solution of the dynamic system equations of physics have been studied by using algebraic and numerical methods. In recent times Lie algebraic techniques have been used to construct complex quasi exactly solvable potentials with real spectrum. In our study, we find the solution of the several different Schrödinger equations using the method of Lie algebra and show how the use of Lie algebra helps in simplifying the eigenvalue problem. In the present study, we begin with a specific differential realization of the SU(l,l)«SO(2,l) algebra which can be used to derive the second order differential equation. Then apply variable and similarity transformations to the group generators in order to recover the Schrödinger equations for various potentials. We also demonstrated that non-Hermition PT symmetric Hamiltonians have real eigenvalues. In addition, the two well-known non linear equations, the Heat and Lorenz equations are solved by numerical Runge Kutta 4 method and via Mathematica in physics. The effects of parameters and the initial conditions are examined. Finally, we review the topological analysis of data generated by a dynamical system operating a chaotic regime. UlKey Words: Lie Algebra, Realization of SO(2,l), PT Symmetric Hamiltonians and Dynamical System. IV