A free boundary problem for the localization of eigenfunctions

We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schr\"odinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.Comment: 174 page

Diffusion-Reorganized Aggregates: Attractors in Diffusion Processes?

A process based on particle evaporation, diffusion and redeposition is applied iteratively to a two-dimensional object of arbitrary shape. The evolution spontaneously transforms the object morphology, converging to branched structures. Independently of initial geometry, the structures found after long time present fractal geometry with a fractal dimension around 1.75. The final morphology, which constantly evolves in time, can be considered as the dynamic attractor of this evaporation-diffusion-redeposition operator. The ensemble of these fractal shapes can be considered to be the {\em dynamical equilibrium} geometry of a diffusion controlled self-transformation process.Comment: 4 pages, 5 figure

Localization landscape theory of disorder in semiconductors I: Theory and modeling

We present here a model of carrier distribution and transport in semiconductor alloys accounting for quantum localization effects in disordered materials. This model is based on the recent development of a mathematical theory of quantum localization which introduces for each type of carrier a spatial function called \emph{localization landscape}. These landscapes allow us to predict the localization regions of electron and hole quantum states, their corresponding energies, and the local densities of states. We show how the various outputs of these landscapes can be directly implemented into a drift-diffusion model of carrier transport and into the calculation of absorption/emission transitions. This creates a new computational model which accounts for disorder localization effects while also capturing two major effects of quantum mechanics, namely the reduction of barrier height (tunneling effect), and the raising of energy ground states (quantum confinement effect), without having to solve the Schr\"odinger equation. Finally, this model is applied to several one-dimensional structures such as single quantum wells, ordered and disordered superlattices, or multi-quantum wells, where comparisons with exact Schr\"odinger calculations demonstrate the excellent accuracy of the approximation provided by the landscape theory.Comment: 17 pages, 18 figures, 3 table

Effective Confining Potential of Quantum States in Disordered Media

The amplitude of localized quantum states in random or disordered media may exhibit long-range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long-range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that Weyl’s formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.National Science Foundation (U.S.) (Grant DMS-1069225)National Science Foundation (U.S.) (Grant DMS-1500771

Optimal branching asymmetry of hydrodynamic pulsatile trees

Most of the studies on optimal transport are done for steady state regime conditions. Yet, there exists numerous examples in living systems where supply tree networks have to deliver products in a limited time due to the pulsatile character of the flow. This is the case for mammals respiration for which air has to reach the gas exchange units before the start of expiration. We report here that introducing a systematic branching asymmetry allows to reduce the average delivery time of the products. It simultaneously increases its robustness against the unevitable variability of sizes related to morphogenesis. We then apply this approach to the human tracheobronchial tree. We show that in this case all extremities are supplied with fresh air, provided that the asymmetry is smaller than a critical threshold which happens to fit with the asymmetry measured in the human lung. This could indicate that the structure is adjusted at the maximum asymmetry level that allows to feed all terminal units with fresh air.Comment: 4 pages, 4 figure

One single static measurement predicts wave localization in complex structures

A recent theoretical breakthrough has brought a new tool, called \emph{localization landscape}, to predict the localization regions of vibration modes in complex or disordered systems. Here, we report on the first experiment which measures the localization landscape and demonstrates its predictive power. Holographic measurement of the static deformation under uniform load of a thin plate with complex geometry provides direct access to the landscape function. When put in vibration, this system shows modes precisely confined within the sub-regions delineated by the landscape function. Also the maxima of this function match the measured eigenfrequencies, while the minima of the valley network gives the frequencies at which modes become extended. This approach fully characterizes the low frequency spectrum of a complex structure from a single static measurement. It paves the way to the control and engineering of eigenmodes in any vibratory system, especially where a structural or microscopic description is not accessible.Comment: 5 pages, 4 figure

Localization landscape theory of disorder in semiconductors. III. Application to carrier transport and recombination in light emitting diodes

This paper introduces a novel method to account for quantum disorder effects into the classical drift-diffusion model of semiconductor transport through the localization landscape theory. Quantum confinement and quantum tunneling in the disordered system change dramatically the energy barriers acting on the perpendicular transport of heterostructures. In addition they lead to percolative transport through paths of minimal energy in the 2D landscape of disordered energies of multiple 2D quantum wells. This model solves the carrier dynamics with quantum effects self-consistently and provides a computationally much faster solver when compared with the Schr\"odinger equation resolution. The theory also provides a good approximation to the density of states for the disordered system over the full range of energies required to account for transport at room-temperature. The current-voltage characteristics modeled by 3-D simulation of a full nitride-based light-emitting diode (LED) structure with compositional material fluctuations closely match the experimental behavior of high quality blue LEDs. The model allows also a fine analysis of the quantum effects involved in carrier transport through such complex heterostructures. Finally, details of carrier population and recombination in the different quantum wells are given.Comment: 14 pages, 16 figures, 6 table

The landscape law for the integrated density of states

The present paper establishes non-asymptotic estimates from above and below on the integrated density of states of the Schr\"odinger operator $L=-\Delta+V$, using a counting function for the minima of the localization landscape, a solution to the equation $Lu=1$