  In the standard formulation of quantum mechanics, one starts by proposing a
potential function that models the physical system. The potential is then
inserted into the Schr\"odinger equation, which is solved for the wave
function, bound states energy spectrum and/or scattering phase shift. In this
work, however, we propose an alternative formulation in which the potential
function does not appear. The aim is to obtain a set of analytically realizable
systems, which is larger than in the standard formulation and may or may not be
associated with any given or previously known potential functions. We start
with the wavefunction, which is written as a bounded infinite sum of elements
of a complete basis with polynomial coefficients that are orthogonal on an
appropriate domain in the energy space. Using the asymptotic properties of
these polynomials, we obtain the scattering phase shift, bound states and
resonances. This formulation enables one to handle not only the well-known
quantum systems but also previously untreated ones. Illustrative examples are
given for two- and there-parameter systems.
