Dispersive estimates for Schr\"odinger operators with point interactions in R3\mathbb{R}^3

The study of dispersive properties of Schr\"odinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schr\"odinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p-L^q$ estimates of the free Laplacian in the smaller regime $q\in[2,3)$. These estimates are implied by a recent result concerning the $L^p$ boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q\geq 3$.Comment: To appear on: "Advances in Quantum Mechanics: Contemporary Trends and Open Problems", G. Dell'Antonio and A. Michelangeli eds., Springer-INdAM series 201

Modelling of Quantum Dots for Intermediate Band Solar Cells

We present a theoretical model for design and analysis of semiconductor quantum dot (QD) array-based intermediate band solar cell (IBSC). The plane wave method with periodic boundary conditions is used in expansion of the k⋅p Hamiltonian for calculation of the electronic and optical structure of InAs/GaAs QD array. Taking into account realistic QD shape, QD periodicity in the array, as well as effects like band mixing between states in the conduction and valence band, strain and piezoelectric field, the model reveals the origin of the intermediate band formation inside forbidden energy gap of the barrier material. Having established the interrelation between QD periodicity and the electronic structure across the QD array Brillouin zone, conditions are identified for the appearance of pure zero density of states regions that separate intermediate band from the rest of the conduction band. For one realistic QD array, we have estimated all important absorption spectra in IBSC, and most important, radiative and nonradiative scattering times. Under radiative limit approximation, we have estimated efficiency of such IBSC to be 39%.Part of the Springer Series in Optical Sciences book series (SSOS,volume 165)