Related papers: Carleman estimates and absence of embedded eigenva…

In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some…

We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schr\"{o}dinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal.…

We consider non-self-adjoint Schr\"{o}dinger operators $H_{{\rm c}}=-\Delta+V_{{\rm c}}$ (resp. $H_{{\rm r}}=-\Delta+V_{{\rm r}}$) acting in $L^2(\mathbb R^d)$, $d\ge 1$, with dilation analytic complex (resp. real) potentials. We were able…

We derive a sharp bound on the location of non-positive eigenvalues of Schroedinger operators on the halfline with complex-valued potentials.

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As…

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this…

We utilize the Bellman function technique to prove a bilinear dimension-free inequality for the Hermite operator. The Bellman technique is applied here to a non-local operator, which at first did not seem to be feasible. As a consequence of…

In this paper, we first improve some asymptotic formulas previously obtained and provide sharp asymptotic formulas explicitly expressed by the potential. For the potentials of bounded variation, we obtain asymptotic formulas in which the…

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave…

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…

This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known.…

We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition,…

The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer $L^p$ spaces for the wave packet transform of functions in $ L^p(\mathbb R)$, in the $2\leq p\leq \infty$ range referred to as local $L^2$. In…

A complex number $\lambda$ is called an extended eigenvalue of a bounded linear operator $T$ on a Banach space $\B$ if there exists a non-zero bounded linear operator $X$ acting on $\B$ such that $XT=\lambda TX$. We show that there are…

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy…

Estimates for eigenvalues of Schr\"{o}dinger operators on the half-line with complex-valued potentials are established. Schr\"{o}dinger operators with potentials belonging to weak Lebesque's classes are also considered. The results cover…

In this paper, by constructing the weight functions, a global Carleman estimate for the Schrodinger equation on a tree is established, with a strong assumption on the solution. And the estimate is able to be applied to derive the Lipschitz…

Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schr\"odinger operators and control theory. We review recent results and announce new ones regarding…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain…

Suppose that $\Sigma^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $\lambda_1$ of the induced Laplace-Beltrami operator on $\Sigma$ satisfies $\lambda_1 \geq \frac{n}{2}+…