  A new solvable hyperbolic single wave potential is found by expanding the
regular solution of the 1D Schr\"odinger equation in terms of square integrable
basis. The main characteristic of the basis is in supporting an infinite
tridiagonal matrix representation of the wave operator. However, the
eigen-energies associated with this potential cannot be obtained using
traditional procedures. Hence, a new approach (the "potential parameter"
approach) has been adopted for this eigenvalue problem. For a fixed energy, the
problem is solvable for a set of values of the potential parameters (the
"parameter spectrum"). Subsequently, the map that associates the parameter
spectrum with the energy is inverted to give the energy spectrum. The bound
states wavefunction is written as a convergent series involving products of the
ultra-spherical Gegenbauer polynomial in space and a new polynomial in energy,
which is a special case of the "dipole polynomial" of the second kind.
