Etude du hamiltonien matriciel d’ordre deux dépendants du potentiel elliptique de Jacobi partiellement algébrique

Abstract

The study of partially algebraic hamiltonian operators occupies a central place in quantum mechanics. The main objective is to determine the energiy spectrum of a quantum system. Our work focuses on the construction of partially algebraic hamiltonian operator and subsequently on the determination of the necessary and sufficient conditions for the secondorder matrix Hamiltonian dependent on the Jacobi elliptic potential to be partially algebraic. The first chapter covers the generalities and basic concepts of partially algebraic Hamiltonians. In the second chapter, using the analytical method of partial solvability, I highlighted the three necessary and sufficient conditions for the second-order matrix Hamiltonian dependent on the Jacobi elliptic potential to be partially algebraic for case 𝛿 = 1. In the third and final chapter, using the same method employed in the previous chapter, I proved that the Hamiltonian described above is partially algebraic, this time for case 𝛿 = 2.