Existence of the Stark-Wannier quantum resonances

In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.Comment: 30 pages, 1 figur

TernausNetV2: Fully Convolutional Network for Instance Segmentation

The most common approaches to instance segmentation are complex and use two-stage networks with object proposals, conditional random-fields, template matching or recurrent neural networks. In this work we present TernausNetV2 - a simple fully convolutional network that allows extracting objects from a high-resolution satellite imagery on an instance level. The network has popular encoder-decoder type of architecture with skip connections but has a few essential modifications that allows using for semantic as well as for instance segmentation tasks. This approach is universal and allows to extend any network that has been successfully applied for semantic segmentation to perform instance segmentation task. In addition, we generalize network encoder that was pre-trained for RGB images to use additional input channels. It makes possible to use transfer learning from visual to a wider spectral range. For DeepGlobe-CVPR 2018 building detection sub-challenge, based on public leaderboard score, our approach shows superior performance in comparison to other methods. The source code corresponding pre-trained weights are publicly available at https://github.com/ternaus/TernausNetV

On spectral stability of solitary waves of nonlinear Dirac equation on a line

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.Comment: 20 pages with figure

Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized nonlinear Schr\"odinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.Comment: latex2e, 41 pages, 1 figur