8 problems

Quantum-layer discrete-spectrum conjecture. If $\varSigma$ is not the plane and $K$ is integrable on $\varSigma$, then the Dirichlet Laplacian on $\Omega$ has non-empty discrete sp…

Let $\varepsilon$ denote the window parameter, let $\lambda_\varepsilon$ be the emerging eigenvalue, and let $c$ be a constant. Let the non-trivial solution of equation (1.9) satis…

Let $\mathfrak{i}_1$ denote the quantity associated with the unique bounded non-trivial solution of equation (1.9), and suppose that this solution exists uniquely. An eigenvalue is…

Positive-curvature conjecture.

Let $\varSigma$ be a complete surface in $\mathbb R^3$ satisfying the stated hypotheses, and let $\kappa$ be its Gauss curvature. Assume $\kappa$ satisfies the stated integrability…

Let $\varSigma$ be a surface satisfying the hypotheses on the embedded quantum layer, and let $B_\infty$ denote the supremum of the norm of its second fundamental form. Assume the…

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…

Let $p:\varSigma\to\mathbb R^3$ be an embedded, orientable, complete, non-compact surface that is not totally geodesic and whose second fundamental form tends to zero at infinity.…