Clusters of Neumann--Poincar\'e eigenvalues need not sum to one half on perturbed spheres

The Neumann--Poincar\'e operator $\mathcal K^*$ of a closed surface is the boundary integral operator for which $\pm\frac12 I+\mathcal K^*$ gives the normal derivatives, from outside and inside, of the single layer potential $-\int\xi(y)\,dS_y/(4\pi|x-y|)$ of a density $\xi$ on the surface. On the unit sphere, its eigenvalues are $1/(2(2k+1))$, $k=0,1,2,\dots$, each with multiplicity $2k+1$, so for every $k$ these $2k+1$ eigenvalues sum to $1/2$. On a perturbed sphere, $1/(2(2k+1))$ splits into a cluster of $2k+1$ eigenvalues. Ando, Kang, Miyanishi and Ushikoshi posed the 1/2-problem, whether the sum of this cluster stays $1/2$, and proved that the first variation of the sum vanishes. We therefore compute the second-order term of the sum $S_k(\eps)$ on the surfaces $r=1+\eps P_l(\cos\theta)$, where $P_l$ is the polynomial of degree $l$ with $P_l(1)=1$ that is orthogonal on $[-1,1]$ to every polynomial of lower degree. These surfaces are symmetric about the axis $\theta=0$, so the eigenvalue problem splits into subspaces of functions that change by a fixed factor $e^{im\phi}$ under the rotation by $\phi$ about this axis, and each subspace holds at most one eigenvalue of the cluster. We expand that eigenvalue to second order in $\eps$ and prove the expansion correct up to $O(\eps^3)$. For $l=3$, the coefficient of $\eps^2$ in $S_k(\eps)$ is a rational function of $k$ that is positive for every $k\ge1$, so for each $k\ge1$, $S_k(\eps)>1/2$ for every small enough $\eps\ne0$, and the 1/2-problem has a negative answer. For $l=2k-1$, the coefficient is negative for $3\le k\le20$, the range we computed, so the sum can also fall below $1/2$.