  The Schr\"odinger equation with a Lennard-Jones potential is solved by using
a procedure that treats in a rigorous way the irregular singularities at the
origin and at infinity. Global solutions are obtained thanks to the computation
of the connection factors between Floquet and Thom\'e solutions. The energies
of the bound states result as zeros of a function defined by a convergent
series whose successive terms are calculated by means of recurrence relations.
The procedure gives also the wave functions expressed either as a linear
combination of two Laurent expansions, at moderate distances, or as an
asymptotic expansion, near the singular points. A table of the critical
intensities of the potential, for which a new bound state (of zero energy)
appears, is also given.
