Remarks on an eigenvalue problem associated with the periodic sine-Gordon equation

The time flow of the periodic Sine-Gordon equation, q/sub tt/ - q/sub xx/ + sin q = 0 fixes the periodic and antiperiodic spectrum of a certain differential operator Q with periodic coefficients (q,p) where p = q/sub t/. The isospectral class L(q0,p0) consisting of all coefficients (q,p) with the same periodic and antiperiodic spectrum as a given coefficient (q0,p0) is studied in detail. Compactness and regularity results for L(q0,p0) are proven by means of various eigenfunction identities. An example is given which shows that L(q0,p0) need not be connected and another example is given which shows that L(q0,p0) may have singularities. A particle system is developed which reduces the inverse problem to solution of a differential equation. This thesis extends the work of McKean and Trubowitz on Hill's equation to the sine-Gordon equation