Lie Symmetries of Schr¨odinger Equation on a Sphere

We are going to discuss an object with a mass attached to a spring and vibrating on the surface of a sphere (see figure 1). to do this, we first reconstruct the schr¨odinger equation on a sphere. in fact, the paper considers the question of a quantum system obeying the schrodinger equation on a sphere. after a brief introduction we set up the hamiltonian of the system and the corresponding schrodinger equation. we provide the lie algebra of symmetries and build the optimal system of lie subalgebras and its adjoint presentation. reductions of similarities related to subalgebras are obtained which are used in our study of the 3d quantum harmonic oscillator on a sphere as a special case of the new equation, and possible solutions are proposed.