7 problems

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…

Consider a family of leaky-graph Hamiltonians in the strong-coupling regime, with energy reference point chosen as $\lambda\alpha^2$ and with the stated condition on $\lambda$. Lea…

Let $\Gamma$ be otherwise smooth but have one angle equal to $\pi-\beta$, and let $\lambda_j$ be the $j$-th eigenvalue of the corresponding leaky star-graph Hamiltonian…

Let $\Gamma$ be a connected infinite $C^4$-smooth curve that is straight outside a compact set. Let $\mathcal{S}_{\Gamma,\alpha}(\lambda)$ be the on-shell scattering matrix of…

Let $\Gamma$ be an $N$-armed star graph with sector angles $\beta_1,\ldots,\beta_N$ satisfying the geometric constraints for a star, and let the ground-state energy of the correspo…

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…

Let $\Gamma\subset\mathbb{R}^3$ be a single smooth curve, and suppose a family of scaled potential wells in the normal plane has a zero-energy resonance, as in approximations of a…