A key basis for seeking periodic solutions of the Camassa-Holm equation is to understand the associated spectral problem The periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum. The potential can then be reconstructed from the periodic spectrum. A necessary and sufficient condition for exponential decrease of the widthsλ2n−λ2n−1for a sequence 0<λ1⩽λ2<… of single or double eigenvalues tending to infinity is the real analyticity ofm. The case of a purely simple spectrum is typical of 0>m∈C1(R).