Two-sided eigenvalue bounds for the spherically symmetric states of the Schrِdinger equation

The eigenvalues of the radial Schrödinger equation are calculated very accurately by obtaining exact upper and lower bounds. By truncating the usual unbounded domain [0, ∞) of the system to a finite interval of the form [0,l], two auxiliary eigenvalue problems are defined. It is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required. Moreover, each auxiliary eigenvalue problem gives rise to an orthonormal set involving Bessel functions. The matrix representation of the Hamiltonian is, therefore, derived by expanding the wave function into a Fourier-Bessel series. Numerical results for single- and double-well polynomial oscillators as well as Gaussian type non-polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy, whenever the boundary parameter l is in the neighborhood of some critical value, denoted by lcr.