相关论文: Quantitative uniqueness for zero-order perturbatio…

We give explicit analytic criteria for two problems associated with the Schr\"odinger operator $H = -\Delta + Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and…

Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The…

We prove three theorems about the asymptotic behavior of solutions $u$ to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index…

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the…

We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded.…

We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus…

The purpose of this note is to investigate the high frequency behaviour of solutions to linear Schr\"odinger equations. More precisely, Bourgain and Anantharaman-Macia proved that any weak-* limit of the square density of solutions to the…

We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials…

The problem of giving a (CR-)geometric description of the best possible order of a subelliptic estimate at a boundary point in the $\bar\partial$-Neumann problem is largely open. In this paper, we introduce a novel technique based on a…

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we establish an optimal regularity result for…

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schr\"odinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on…

We establish sharp boundary regularity results for solutions to kinetic Fokker-Planck equations under prescribed inflow boundary conditions, providing precise quantification of the boundary hypoelliptic regularization effect. For equations…

The asymptotic behavior of solutions to Schr\"odinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the…

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schr\"odinger operator has a finite or infinite number of the discrete pectrum. In…

We establish Lieb-Thirring type inequalities for non self-adjoint relatively compact perturbations of certain operators of mathematical physics. We apply our results to quantum Hamiltonians of Schr{\"o}dinger and Pauli with constant…

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-\theta}(r^{\alpha}\vert…

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a…

We show that formal Schr\"odinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the…