The fundamental gap conjecture and log-concavity

At the end of his study [J. Statist. Phys., 83] on thermodynamic functions of a free boson gas, van den Berg conjectured that the (fundamental) gap $\Gamma^{V}(\Omega):=\lambda_{2}^{V}(\Omega)-\lambda_{1}^{V}(\Omega)$ between the two smallest eigenvalues $\lambda_{1}^{V}(\Omega)$, $\lambda_{2}^{V}(\Omega)$ of the Schrödinger operator $-\Delta + V$ on a convex domain $\Omega$ in $\mathbb{R}^d$, $d\ge 1$, equipped with homogeneous Dirichlet boundary conditions satisfies
\begin{equation} \label{gap} \Gamma^{V}(\Omega)\ge \Gamma\left(I_D \right)=\frac{3\pi^{2}}{D^2}, \end{equation}
where $I_D$ is the interval $(-D/2,D/2)$ of length $D=\text{diameter}(\Omega)$ . Since term $\Gamma^{V}(\Omega)$ measures, for example, the energy needed to jump from the ground state to the next excited eigenstate, one is interested in (optimal) lower bounds on $\Gamma^{V}(\Omega)$. Since the late 80s, the fundamental conjecture \eqref{gap} attracts consistently the attention of many researchers including M. S. Ashbaugh & R. Benguria [Proc. Amer. Math. Soc., 89], R. Schoen and S.-T. Yau [Camb. Press, 94.] (see also [Geneva, 86]), and was finally proven by B. Andrews and J. Clutterbuck [J. Amer. Math. Soc., 11].\medskip
A key result in the proofs of the fundamental gap conjecture is that the first eigenfunction of Schr\”odinger operator is log-concave. Now, on a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this talk, I show that this is false by analyzing the perturbation problem from the Neumann case. First, I classify all convex polyhedral domains on which the first variation of the ground state with respect to the Robin parameter at zero is not a concave function. Then, I conclude from this that the Robin ground state is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on convex domains with smooth boundary which approximate these in Hausdorff distance.