Variational principle for Hamiltonians with degenerate bottom

We consider perturbations of Hamiltonians whose Fourier symbol attains its minimum along a hypersurface. Such operators arise in several domains, like spintronics, theory of supercondictivity, or theory of superfluidity. Variational estimates for the number of eigenvalues below the essential spectrum in terms of the perturbation potential are provided. In particular, we provide an elementary proof that negative potentials lead to an infinite discrete spectrum.Comment: 9 page

On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

Let $\Omega\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $\alpha>0$, let $H^\Omega_\alpha$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary conditions $\partial u/\partial\nu =\alpha u$, where $\nu$ is the exterior unit normal at the boundary of $\Omega$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^\Omega_m(\alpha)$ of $H^\Omega_\alpha$ behaves as E^\Omega_m(\alpha)=-\alpha^2+\mu^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrt\alpha}\Big) \quad {as $\alpha$ tends to $+\infty$}, where $\mu^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f"$ on $(0,\ell_n)$ with the Dirichlet boundary conditions.Comment: 10 pages. To appear in Nanosystems: Physics, Chemistry, Mathematics. Minor revision: misprints corrected, references update

Resolvents of self-adjoint extensions with mixed boundary conditions

We prove a variant of Krein's resolvent formula expressing the resolvents of self-adjoint extensions through the associated boundary conditions. Applications to solvable quantum-mechanical problems are discussed.Comment: 15 pages, rewritten almost completel

An inequality for the maximum curvature through a geometric flow

We provide a new proof of the following inequality: the maximum curvature $k_\mathrm{max}$ and the enclosed area $A$ of a smooth Jordan curve satisfy $k_\mathrm{max}\ge \sqrt{\pi/A}$. The feature of our proof is the use of the curve shortening flow.Comment: 2 page