Exploration of quasi exactly solvable matrix models for Jahn Teller systems

Lie algebras play important role for solving the quantum mechanical problems. Exactly and quasi exactly solvable (QES) systems can be classified using algebraic approachs in which the Hamiltonians are expressed in terms of the generators of the Lie algebra. In this thesis, quasi exact solvability of some Jahn Teller (JT) Hamiltonians and quantum mechanical potentials have been discussed in the framework of the Lie algebra. This work includes two main parts. In the first part, QES quantum mechanical potentials whose eigenstates and eigenfunctions have been obtained in terms of the orthogonal polynomials and these potentials have been obtained from each other by a proper transformation. The second part covers the solutions of the E ⊗ ² and T ⊗ t JT Hamiltonians. It is well known that JT effect (electron vibron coupling) is important phenomenon in high symmetry systems for understanding the structure and dynamics of molecules and solids. The quasi exact solution of the E ⊗ ² JT Hamiltonian has been obtained in the context of the osp(2,2) Lie super algebra.This Hamiltonian has been reduced in the form of one variable differential equations by introducing a similarity transformation technique. The T ⊗t JT Hamiltonian has been solved using the rotating wave approximation method which is based on Lie type transformation and simplifies the mathematical complexity of the JT problems. Its eigenvalues and associated eigenfunctions have been obtained in a closed form when the coupling constant is smaller than the natural frequency of the oscillator. The applicability of this method is firstly demostrated for a simple system known as Jaynes Cumming (JC) Hamiltonian in the literature. These methods can also be extended to solve the other JT Hamiltonians which includes multi boson or multi boson-fermion systems. Key words: Jahn Teller Effect, Lie (super)algebra, Quasi-exact solvability