Solution of the Schrödinger equation for bound states in closed form

A method to calculate the bound-state eigenvalues of the Schr\"odinger equation is presented. The method uses a new diagonal representation of the Hamiltonian. The variational principle is applied to this diagonal representation and yields closed-form expressions of the form ${E}_{n}=E(an+b)$ for the eigenvalues. Examples are presented for some quark potentials of current interest.