Related papers: Weighted inequalities for Schr\"odinger type Singu…

Schr\"odinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically…

We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher…

Let $\Omega$ be a homogeneous function of degree zero and enjoy the vanishing condition on the unit sphere $\mathbb{S}^{n-1}(n\geq 2)$. Let $T_{\Omega}$ be the convolution singular integral operator with kernel ${\Omega(x)}{|x|^{-n}}$. In…

Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these…

We study the $L^\infty(\mathbb{R}^d)$ boundedness for Riesz transforms of the form ${V^{a}(-\frac12\Delta+V)^{-a}},$ where $a > 0$ and $V$ is a non-negative potential with power growth acting independently on each coordinate. We factorize…

It is shown that Schroedinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points…

We study the two dimensional Schr\"odinger operator, $H=-\Delta+V$, in the weighted L^1(\R^2) \rightarrow L^{\infty}(\R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\les \la x…

For two dimensional Schr\"odinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schr\"odinger operator, are bounded in the Lebesgue space $L^p(\R^2)$ for…

We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an…

The spectrum of discrete Schr\"odinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the…

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schr\"odinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-\delta}$ for some $\delta>2$ when $H$ is of exceptional…

We study mapping properties of two-dimensional linear integral operators in some weighted spaces with special kernels. The considered spaces are certain variant of Sobolev--Slobodetskii spaces and their generalizations related to Banach…

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_\Omega$ associated with an integrable function…

By $\{T_t^a\}_{t>0}$ we denote the semigroup of operators generated by the Friedrichs extension of the Schr\"odinger operator with the inverse square potential $L_a=-\Delta+\frac{a}{|x|^2}$ defined in the space of smooth functions with…

We prove resolvent estimates in the Euclidean setting for Schr\"{o}dinger operators with potentials in Lebesgue spaces: $-\Delta+V$. The $(L^{2}, L^{p})$ estimates were already obtained by Blair-Sire-Sogge, but we extend their result to…

Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older…

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the…

Bounded and unbounded weighted composition operators on $L^2$ spaces over $\sigma$-finite measure spaces are investigated. A variety of questions related to seminormality of such operators are discussed.

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away…

Consider the Schr\"odinger operator ${\cal A}=-\frac{\Delta}{2}+V$ acting on space $C_0^\infty(D)$, where $D$ is an open domain in $\R^d$. The main purpose of this paper is to present the $L^\infty(D,dx)$-uniqueness for Schr\"odinger…