Quantum-layer bound-state conjecture for integrable Gauss curvature

Let $\varSigma$ be a complete, noncompact surface embedded in $\mathbb R^3$ satisfying the preceding hypotheses, and let $\kappa$ be its Gauss curvature. Assume

$$\int_\varSigma |\kappa|\,d\varSigma<+\infty.$$

Quantum-layer bound-state conjecture. There exists $\alpha=\alpha(\varSigma)$ such that, for every $a\in(0,\alpha)$, the discrete spectrum of the quantum layer over $\varSigma$ of width $2a$ is non-empty. This conjecture was proved under the condition $\int_\varSigma\kappa\,d\varSigma\leq0$ through work of Duclos, Exner and Krejčiřík and of Carron, Exner and Krejčiřík; the remaining positive-total-curvature case is addressed by later results in the paper.