Related papers: Semiclassical resolvent bounds in dimension two

In dimension two, we investigate a free energy and the ground state energy of the Schr\"odinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling…

We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove…

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent…

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 deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_\alpha ^m$, from a product of weighted Lebesgue spaces into adequate…

It is well known that, with a particular choice of norm, the classical double-layer potential operator $D$ has essential norm $<1/2$ as an operator on the natural trace space $H^{1/2}(\Gamma)$ whenever $\Gamma$ is the boundary of a bounded…

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$,…

In this paper, we study the following Schr\"odinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ & \int |u|^2dx=\Theta, \end{aligned}…

For real functions \Phi and \Psi that are integrable and compactly supported, we prove the norm resolvent convergence, as \epsilon\ goes to 0, of a family S(\epsilon) of one-dimensional Schroedinger operators on the line of the form…

We prove the existence of ballistic transport for the Schr\"odinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior…

We study the semi-classical trace formula at a critical energy level for a Schr\"odinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result,…

Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment…

We prove polynomial bounds on the growth of higher Sobolev norms for the nonlinear Schrodinger equation set on a torus, in dimension 3, with super-cubic and sub-quintic nonlinearity. Due to improved Strichartz estimates, these bounds are…

The supersymmetric approach in the form of second order intertwining relations is used to prove the exact solvability of two-dimensional Schrodinger equation with generalized two-dimensional Morse potential for $a_0=-1/2$. This…

We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…

We consider semiclassical Schroedinger operators on R^n, with C^\infty potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we…

We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to…

Generalizing previous results obtained for the spectrum of the Dirichlet and Neumann realizations in a bounded domain of a Schr\"odinger operator with a purely imaginary potential $h^2\Delta+iV$ in the semiclassical limit $h\to 0$ we…

We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive…

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $\sigma\geq 3/2$ established in an article by…