Numerical determination of bound states without matrix diagonalization

We present a simply applied numerical technique that allows the accurate determination of the bound-state eigenfunctions and eigenvalues of a differential operator such as the one-particle Schrödinger Hamiltonian. The method applies for potentials that asymptotically vanish. The eigenvalues and eigenfunctions are determined as functions of the strength of the potential and the method is able to determine the bound-state energies for arbitrarily weak strengths of the potential. At no point is a matrix diagonalized thus the method may be applied to problems with space dimension greater than unity.