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

Abstract

The Weinstein equation with complex coefficients is the equation governing generalized axisymmetric potentials (GASP) which can be written as $L_m[u]=\Delta u+\left(m/x\right)\partial_x u =0$, where $m\in\mathbb{C}$. We generalize results known for $m\in\mathbb{R}$ to $m\in\mathbb{C}$. 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 $\mathbb{H}^+$ for Re $m<1$. We establish a new decomposition theorem for the GASP in any annular domains for $m\in\mathbb{C}$, which is in fact a generalization of the B\^ocher'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 $m\in\mathbb{C}$, we show that this family is even a Riesz basis in some non-concentric circular annulus.