Optimization of the lowest eigenvalue of a soft quantum ring

We consider the self-adjoint two-dimensional Schr\"odinger operator $H_\mu$ associated with the differential expression $-\Delta -\mu$ describing a particle exposed to an attractive interaction given by a measure $\mu$ supported in a closed curvilinear strip and having fixed transversal one-dimensional profile measure $\mu_\bot$. This operator has nonempty negative discrete spectrum and we obtain two optimization results for its lowest eigenvalue. For the first one, we fix $\mu_\bot$ and maximize the lowest eigenvalue with respect to shape of the curvilinear strip the optimizer in the first problem turns out to be the annulus. We also generalize this result to the situation which involves an additional perturbation of $H_\mu$ in the form of a positive multiple of the characteristic function of the domain surrounded by the curvilinear strip. Secondly, we fix the shape of the curvilinear strip and minimize the lowest eigenvalue with respect to variation of $\mu_\bot$, under the constraint that the total profile measure $\alpha >0$ is fixed. The optimizer in this problem is $\mu_\bot$ given by the product of $\alpha$ and the Dirac $\delta$-function supported at an optimal position.