Decomposition theorem and Riesz basis for axisymmetric potenials in the right hal-plane

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The Weinstein equation with complex coefficients is the equation governing generalized axisymmetric potentials (GASP) which can be written as Lₘ[u]=Δu+(m/x)∂ₓ u =0, where m. We generalize results known for m to m. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for GASP in the right half-plane H+ for Re m<1. We establish a new decomposition theorem for the GASP in any annular domains for m, which is in fact a generalization of the Bôcher's decomposition theorem. In particular, using bipolar coordinates, we prove for annuli that a family of solutions for GASP equation in terms of associated Legendre functions of first and second kind is complete. For m, we show that this family is even a Riesz basis in some non-concentric circular annulus.