A precise description of the location of the spectrum

The intervals of one of the families that do not intersect any interval of the other family
are called non-resonant, the others being the resonant intervals.
We mainly concentrate on the non resonant case.

One may wonder if resonances occur.
One may wonder if resonances occur. They do occur. 

Indeed, the derivatives of  and  are of opposite signs
Indeed, the derivatives of  and  are of opposite signs and the energies are given by

Hence, as  decreases, the points of type  and  move toward each other; so, they meet.

Nevertheless, for a generic , there are only a few resonant intervals.
On the other hand, for symmetric , there may be numerous resonant energies; e.g.,
if  is even, then the two sequences coincide and all the intervals are resonant!

We describe the results for the intervals of type ; for the other type, the results are
analogous.