A covering-space-like approach to a semi-differential equation

Dewar and Glasser famously treated ballooning modes using a covering space: They eased treatment of the parallel gradient by solving on an infinite plane, rather than a periodic torus. They reconstructed the true, periodic solution via linear recombination of degenerate solutions on the plane. We stumbled on an analogous problem in a different context: steady-state diffusion in an axisymmetric torus with an ideal limiter. The poloidal angle is the timelike coordinate. It is periodic in the confined plasma (x<0) and non-periodic outside (x>0). Solutions may be written with semi-differential operators, which iterate to differentiation or integration. The definite semi-integral, appropriate for x>0, iterates to an integral with fixed lower bound; its concrete form appears in the literature. The periodic semi-integral, appropriate for x<0, iterates to the periodic integral, basically dividing each Fourier component by its wave number. We found a real-space form for this operator by expanding the periodic line segment to the full real line, applying the definite semi-integral, then using a modified linear recombination. We are left to wonder: could one apply linear recombination directly to the linear ballooning-mode operator itself, rather than any particular solution?