Related papers: Exponential decay for the Schr{\"o}dinger equation…
Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when…
A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new…
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{\"o}dinger-type equations. These results illustrate the slowdown of propagation…
We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect…
We study the asymptotic behavior of the Euler-Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the "exponential decay" of the memory…
We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schr\"odinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay…
We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial…
We investigate trend to equilibrium for the damped wave equation with a confining potential in the Euclidean space. We provide with necessary and sufficient geometric conditions for the energy to decay exponentially uniformly. The proofs…
We prove unique continuation properties for linear variable coefficient Schr\"odinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any…
We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schr\"odinger equations, perturbed by bounded noise acting on only two Fourier modes. To tackle the lack of smoothing effect, we introduce…
Negative refractive index materials have attracted significant research attention due to their unique electromagnetic response characteristics. In this paper, we employ the complementing boundary condition to establish rigorous a priori…
We consider the spectrum of a Schroedinger operator in a multi-dimensional cylinder perturbed by a shrinking potential. We study the phenomenon of a new eigenvalue emerging from the threshold of the essential spectrum and give the…
We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous,…
We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular…
We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as…
For first order systems, we obtain an efficient bound on the exponential decay of an eigenfunction in terms of the distance between the corresponding eigenvalue and the essential spectrum. As an example, the Dirac operator is considered.
We are concerned with the properties of weak solutions of the stationary Schr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$. Assuming $f$ to…
In this short note, we prove a decay estimate for non-linear solutions of 3D cubic defocusing non-linear Schr\"odinger equation.
The current paper is devoted to the scattering theory of a class of continuum Schr\"{o}dinger operators with deterministic sparse potentials. We first establish the limiting absorption principle for both modified free resolvents and…
We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the…