Related papers: Dynamical self-averaging for a lattice Schr\"oding…

We consider discrete Schr{\"o}dinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre…

This article is concerned with the quasilinear Schr\"odinger equation \[ \Delta u-\omega u+|u|^{p-1}u+\delta\Delta(|u|^2)u=0, \] where $\delta>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and…

We study the linear relaxation Boltzmann equation, a simple semiclassical kinetic model. We provide a resolvent estimate for an associated non-selfadjoint operator as well as an estimate on the return to equilibrium. This is done using a…

Starting from a finitely ramified self-similar set $X$ we can construct an unbounded set $X_{<\infty>}$ by blowing-up the initial set $X$. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace…

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use…

Recently several lattice collaborations have studied the scale dependence of the coupling in theories with different gauge groups and fermion representations using the Schr\"odinger functional method. This has motivated us to look at the…

We study the discrete nonlinear Schr\"oinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved, but on the other,…

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the nonlinear Schr\"odinger equation with wave operator on the torus when the core size of vortex $\varepsilon \to 0$. It is proved that the reduced dynamical…

Let $V_\Gamma$ be a lattice periodic potential and $A$ and $\phi$ external electromagnetic potentials which vary slowly on the scale set by the lattice spacing. It is shown that the Wigner function of a solution of the Schroedinger equation…

We discuss the (in)stability of solitary waves for a quasi-linear Schr{\"o}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the…

Schr\"{o}dinger bridge is a stochastic optimal control problem to steer a given initial state density to another, subject to controlled diffusion and deadline constraints. A popular method to numerically solve the Schr\"{o}dinger bridge…

We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs…

For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, $f$, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and,…

The small dispersion limit of the focusing nonlinear Schrodinger equation with periodic initial conditions is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that…

The spectrum of random ergodic Schr\"odinger-type operators is almost surely a deterministic subset of the real line. The random operator can be considered as a perturbation of a periodic one. As soon as the disorder is switched on via a…

We study Schroedinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate. This is a key ingredient in an existence proof of pure…

Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schr\"odinger operators with localized singular rank-two perturbations coupled with {\delta}-like…

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schr\"odinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schr\"odinger…

We provide a characterization of the spectral minimum for a random Schr\"odinger operator of the form $H=-\Delta + \sum_{i \in \Z^d}q(x-i-\omega_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly…

We build a new estimate for the normalized eigenfunctions of the operator $-\partial_{xx}+\mathcal V(x)$ based on the oscillatory integrals and Langer's turning point method, where $\mathcal V(x)\sim |x|^{2\ell}$ at infinity with $\ell>1$.…