Related papers: Semiclassical Dynamics with Exponentially Small Er…

In this paper we consider linear, time dependent Schr\"odinger equations of the form $i \partial_t \psi = K_0 \psi + V(t) \psi $, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are…

We consider the semiclassical Schr\"odinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2} \Delta + W_l(x) \right)\psi + V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form…

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are…

In this paper we consider linear, time dependent Schr\"odinger equations of the form ${\rm i} \partial_t \psi = K_0 \psi + V(t) \psi$, where $K_0$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral…

We consider semiclassical Schr\"odinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying…

We consider a class of linear time dependent Schr\"odinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schr\"odinger equations on arbitrary flat tori. For the linear Schr\"odinger equation, we prove a…

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_t\psi = ic\hbar\sum_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a…

In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where…

In this paper we consider time dependent Schr{\"o}dinger linear PDEs of the form i$\partial$t$\psi$ = L(t)$\psi$, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for…

In this paper we consider time dependent Schr\"odinger equations on the one-dimensional torus $\T := \R /(2 \pi \Z)$ of the form $\partial_t u = \ii {\cal V}(t)[u]$ where ${\cal V}(t)$ is a time dependent, self-adjoint pseudo-differential…

We consider the dynamics generated by the Schroedinger operator $H=-{1/2}\Delta + V(x) + W(\epsi x)$, where $V$ is a lattice periodic potential and $W$ an external potential which varies slowly on the scale set by the lattice spacing. We…

We find exact solutions of the time-dependent Schr\"odinger equation for a family of quasi-exactly solvable time-dependent potentials by means of non-unitary gauge transformations.

In this paper, we discuss a class of nonlinear Schr\"odinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + \Delta ) \psi = \lambda |\psi|^{2\eta}\psi$ in $\mathbf R^N \times \mathbf R^+$. Based on…

We study solutions of the time dependent Schr\"odinger equation on Riemannian manifolds with oscillatory initial conditions given by Lagrangian states. Semiclassical approximations describe these solutions for small h (where h is the…

We consider the following Schr\"{o}dinger equation $$ - \hslash ^2 \Delta u + V(x)u = \Gamma(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical…

We analyse a class of time discretizations for solving the nonlinear Schr\"odinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show that these…

The time dependent complex Schr\"odinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space…

Time dependent Schr\"odinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations…

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 $\ell>1$.…

We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…