On a particular manifold $\mathcal{M}$, I'm attempting to solve the second-order partial differential equation $$\nabla^2\phi(\mathbf{x}) = \delta(|\mathbf{x}-\mathbf{x}^\prime|)$$ for the unknown potential $\phi$, where $\mathbf{x}^\prime$ is a fixed but arbitrary point in the space, and $\delta$ is the dirac delta function.
The manifold $\mathcal{M}$ is defined by taking the ordinary Euclidean plane $\mathbb{R}^2$, removing two equal-sized discs $D_1,D_2$ and then gluing their boundaries together. For concreteness, we can suppose the discs have radius $a$ and are centered on the $x$-axis at position $x=\pm d$, where $d>a$.
On this manifold $\mathcal{M}$, it is convenient to use bipolar coordinates $\langle \tau, \sigma\rangle$, in which case the boundaries of the two discs are curves of constant $\tau$: $\tau=+\tau_0$ and $\tau=-\tau_0$, and the gluing merges together $\langle \tau_0, \sigma\rangle \sim \langle -\tau_0, \sigma\rangle$
The topology of the space $\mathcal{M}$ imposes boundary conditions on the solution $\phi(\mathbf{x})$, namely:
- At corresponding points on the circular boundaries, the value of the potential $\phi$ must agree.
- At corresponding points on the circular boundaries, the radial derivative of the potential $\partial_\tau \phi$ must agree.
- Very far from the origin (and I think also very near the point charge $\mathbf{x}^\prime$, if it doesn't introduce an inconsistency) the potential should resemble the solution in $\mathbb{R}^2$, namely $\phi_{\mathbb{R}^2}(\mathbf{x}) = \log(|\mathbf{x}-\mathbf{x}^\prime|)$.
My approach has been to guess the general form of the solution $\phi(\mathbf{x})$ and to impose the boundary conditions to determine a specific solution, though I'm getting stuck on the details:
- I thought I could start by decomposing the solution $\phi$ into the fourier modes of $\nabla^2$, since at most points in the space except at $\mathbf{x}^\prime$, the equation is simply $\nabla^2 \phi = 0$. Hence I'd assume $\Phi(\tau,\sigma) \equiv \sum_{k=-\infty}^\infty \phi_k(\tau) \exp(ik\sigma)$ for unknown $\phi_k$.
- Unfortunately, the third boundary condition requires that $\phi$ look like $\log(|\mathbf{x}-\mathbf{x}^\prime|)$ at the extremes, and this is not a square-integrable function and cannot be made as a sum of these modes. So instead, I figured I could separate the solution into a "singular" mode which looks like the solution in $\mathbb{R}^2$, plus nonsingular correction to account for the topology of $\mathcal{M}$: $$\Phi(\tau,\sigma) \equiv \log(|\mathbf{x}-\mathbf{x}^\prime|) + \sum_{k=-\infty}^\infty \phi_k(\tau) \exp(ik\sigma)$$
- I know that bipolar coordinates are conformal, which means that $\nabla^2 \phi(\mathbf{x}) = 0$ just when $\left[ \frac{\partial^2}{\partial\tau^2} + \frac{\partial^2}{\partial\sigma^2}\right]\phi = 0$.
- Conveniently, this forces the fourier modes to satisfy an ordinary differential equation $\phi_k^{\prime\prime} - k^2 \phi_k = 0$, so $\phi_k(\tau)$ is a sum of exponentials (where $k\neq 0$) or linear in $\tau$ (where $k=0$).
- Unfortunately, I don't know how to solve for the remaining two topological boundary conditions to account for the discrepancy $\Phi_{\mathbb{R}^2}(\tau_0, \sigma)-\Phi_{\mathbb{R}^2}(-\tau_0, \sigma)$ and its derivative.
I have considered using the method of images, but in this space, equally-spaced intervals in $\tau$ correspond to very different physical distances in the plane, and the singularity at $\tau=0$ causes additional complications.
(I am not experienced enough with theta functions to know whether they would be applicable to this problem, where the coordinate space $\langle \tau, \sigma\rangle$ is glued together in the shape of a torus, but with a hole at $\tau=0$ where the physical space goes out to infinity.)
Does my general form of solution so far seem plausible? If so, how to proceed?