Algebraic Approach to a Special Four-Body Solvable Model

A special four-body quantum model in one dimension with a discrete spectrum was introduced, including harmonic oscillator and three-body interaction potentials. After reducing one degree of freedom by using the Jacobian transformation in the center of mass, the desired Hamiltonian is examined in spherical coordinate with three degrees of freedom. To investigate this model algebraically, using the gauge rotation with the ground state wave function, the relation between the Hamiltonian and the generators of \(sl(3, {\mathbb {R}})\) $sl(3,\mathbb{R})$ and \(sl(2, {\mathbb {R}})\) $sl(2,\mathbb{R})$ Lie algebras is examined. Finally, this algebraic form helps us to get the Hamiltonian eigenvalues.