Global maximality of the circular leaky loop

Let $H_{\alpha,\Gamma}$ be the planar Hamiltonian with attractive interaction of strength $\alpha>0$ supported by a loop $\Gamma$ of fixed length $L$, and let $\epsilon_1(\alpha,\Gamma)=\inf\sigma(H_{\alpha,\Gamma})$ be its principal eigenvalue. The paper proves that the circle locally sharply maximizes $\epsilon_1(\alpha,\Gamma)$ within the specified class of curves. Global maximality conjecture. The circle is a sharp global maximizer of $\epsilon_1(\alpha,\Gamma)$, even under weaker regularity assumptions. This extends the proved local result to a global isoperimetric statement for leaky loops; the supplied text does not establish the global claim.