相关论文: Sharp convergence for sequences of Schr\"{o}dinger…
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…
In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of…
We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete…
By using the Lie's invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schr\"{o}dinger equations with variable coefficients. Starting from the equivalence generators we construct…
Using Guth's polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schr\"odinger equation in $\mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result…
We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{\"o}dinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law''…
The notion of Fej\'er monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii-Mann iteration, the proximal point method, the Douglas-Rachford splitting algorithm, and many others.…
We consider a non-relativistic quantum particle in $\mathbb{R}^d$, $d=2$ or $d = 3$, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of…
In a real Hilbert space $\mathcal{H}$. Given any function $f$ convex differentiable whose solution set $\argmin_{\mathcal{H}}\,f$ is nonempty, by considering the Proximal Algorithm $x_{k+1}=\text{prox}_{\b_k f}(d x_k)$, where $0<d<1$ and…
We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing…
We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schr\"odinger equation. The generalization yields numerical solutions accurate to order $(\Delta x)^{2r-1}$ in…
For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by…
We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in…
We study non-stationary averaging processes, where each term of a sequence is a weighted average of previous terms, namely $a_{n+1} = \sum_{j=1}^n p_n(j) a_j$. Our results extend classical theory in two distinct regimes. First, we prove a…
For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to…
A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schroedinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansaetze…
We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schr\"odinger equation, both linear and nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the Schr\"odinger…
We improve local smoothing estimates for fractional Schr\"{o}dinger equations for $\alpha \in (0,1) \cup (1,\infty)$.
We show how recent exact results in supersymmetric theories can be extended to models which include {\it explicit} soft supersymmetry breaking terms. We thus derive new exact results for non-supersymmetric models.
We consider Carleson's problem regarding pointwise convergence for the Schr\"odinger equation. Bourgain recently proved that there is initial data, in $H^s(\mathbb{R}^n)$ with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set…