### A model of atomic radiation

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We consider the electronic properties of a system consisting of two quantum dots in physical proximity, which we will refer to as the double-Qdot. Double-Qdots are attractive in light of their potential application to spin-based quantum computing and other electronic applications, e.g. as specialized sensors. Our main goal is to derive the essential properties of the double-Qdot from a model that is rigorous yet numerically tractable, and largely circumvents the complexities of an ab initio simulation....

Nonlinear Schrödinger equations (NLS)${}_{a}$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega $ are considered. If $\Omega ={\mathbb{R}}^{N}$ and $a>-{(N-2)}^{2}/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^{2}/4$ is excluded because $D\left({P}_{a\left(N\right)}^{1/2}\right)$ is not equal to ${H}^{1}\left({\mathbb{R}}^{N}\right)$, where ${P}_{a\left(N\right)}:=-\Delta -{(N-2)}^{2}/{\left(4\right|x|}^{2})$ is nonnegative and selfadjoint in ${L}^{2}\left({\mathbb{R}}^{N}\right)$. On the other hand, if $\Omega $ is a smooth and bounded domain with $0\in \Omega $, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...

In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak$\stackrel{\u02c7}{\mathrm{s}}$ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.