Related papers: Discrete versus continuous wires on quantum networ…
We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis…
For the analysis of the Schr\"odinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space quantitative forms of unique continuation imply Wegner…
We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we…
We investigate uniqueness of solutions to certain classes of elliptic and parabolic equations posed on metric graphs. In particular, we address the linear Schr\"odinger equation with a potential, and the heat equation with a variable…
We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on $L^2(\mathbb{R}^2)$: $H^\lambda_{\rm edge}=-\Delta+\lambda^2…
We elucidate the case in which the Ablowitz-Ladik (AL) type discrete nonlinear Schr\"Aodinger equa- tion (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple…
For the Schr\"odinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and…
An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$…
In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the…
There is great interest in the development of novel nanomachines that use charge, spin, or energy transport, to enable new sensors with unprecedented measurement capabilities. Electrical and thermal transport in these mesoscopic systems…
The solution of gauge theories is one of the most promising applications of quantum technologies. Here, we discuss the approach to the continuum limit for $U(1)$ gauge theories regularized via finite-dimensional Hilbert spaces of quantum…
We consider the three-dimensional cubic nonlinear Schr\"odinger system \begin{equation*} \begin{cases} i\partial_tu+\Delta u+(|u|^2+\beta |v|^2)u=0,\\ i\partial_tv+\Delta v+(|v|^2+\beta |u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be…
We consider the inverse problem for countable, locally finite electrical networks with edge weights in an arbitrary field. The electrical inverse problem seeks to determine the weights of the edges knowing only the potential and current…
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide…
In this paper we study the noncompact star-type graph with perturbed radial Schrodinger equation on each ray and the matching conditions of some special form at the vertex. The results include the uniqueness theorem and constructive…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
Recently it was demonstrated that the concept of a spectral singularity (SS) can be generalized to waves propagating in nonlinear media, like matter waves or electromagnetic waves in Kerr media. The corresponding solutions represent…
Using the path-integral technique we examine the mutual information for the communication channel modelled by the nonlinear Schr\"odinger equation with additive Gaussian noise. The nonlinear Schr\"odinger equation is one of the fundamental…
We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are…
We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schr\"odinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the…