6 problems

Exner's conjecture. For small radii, the bound-state energy depends on $a$ as

Let $\Omega$ be a three-dimensional tube with a gentle, locally deformed cross section $M_x$, and let $H_{m,\Omega}$ be the associated Dirac operator. Suppose that the deformation…

A potential layer is a tubular interaction supported near an infinite surface in $\mathbb{R}^3$, with a fixed transverse channel profile. Its hard-wall counterpart is the correspon…

Let $\Omega$ be the ruled strip with asymptotically diverging twisting considered above, and let $a_1,a_2$ determine its cross-section. Assume that the divergence condition holds a…

The setting is a thin waveguide whose limiting graph has a vertex region, with threshold singularities determined by the associated effective Hamiltonian and influencing the vertex…

Let $H_{\alpha,\Gamma}$ be the leaky-graph Hamiltonian associated with a broken line $\Gamma$ whose angle is parametrized by a small parameter $\theta$, and let $\lambda(\theta)$ d…