Related papers: Vector-valued Schr\"odinger operators on $L^p$-spa…

In this paper, we consider the Schr\"odinger operators $L_k=-\Delta_k+V$, where $\Delta_k$ is the Dunkl-Laplace operator and $V$ is a non-negative potential on $R^d$. We establish that $L_k $ is essentially self-adjoint on $C_0^\infty$. In…

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 study Schr\"odinger operators on $L^2(E;m)$ of the form $-A+V$ with singular potentials $V$. We address the question posed by H. Brezis about the structure of the set $\{u=0\}$ for non-negative supersolutions to $-Au+Vu=0$. The class of…

We study one-dimensional Schr\"{o}dinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class…

Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued…

We extend the Feynman-Kac formula for Schr\"odinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be…

Given $1\leq q<p<\infty$ quantitative weighted L^p estimates, in terms of Aq weights, for vector valued maximal functions, Calder\'on-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular…

Let $\mathcal G$ be a Hilbert space and $\mathfrak B(\mathcal G)$ the algebra of bounded operators, $\mathcal H=L_2([0,\infty);\mathcal G)$. An operator-valued function $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal…

In this paper, we consider nonlocal Schr\"odinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \begin{equation*}L_K u+V u=f\,\,\text{ in $\BR^n$ }\end{equation*} where $V\in\rh^q$ for…

On a Lie group $G$, we investigate the discreteness of the spectrum of Schr\"odinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable function…

Let $L=-\Delta + V(x)$ be a Schr\"odinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz…

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older…

In this work we obtain boundedness results for fractional operators associated with Schr\"odinger operators $\ \mathcal{L}=-\Delta+V$ on weighted variable Lebesgue spaces. These operators include fractional integrals and their respective…

We study Schr\"odinger operators on $\mathbb{R}^2$ $$ H = \left(-\frac{\partial^2}{\partial x_1^2}\right)^{\alpha/2} + \left(-\frac{\partial^2}{\partial x_2^2}\right)^{\alpha/2} + V, $$ for $\alpha \in (0,2)$ and some sufficiently regular,…

Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq3$, where $\Delta$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ for $s\geq d/2$. For…

In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schr\"odinger type operator with unbounded coefficients of the form $$A=a^{2} \Delta ^2+V^{2}$$ where $a(x)=1+|x|^{\alpha}$ and $V=|x|^{\beta}$ with…

We develop the Titchmarsh-Weyl theory for vector-valued discrete Schr\"odinger operators and show that the Weyl $m$ functions associated with these operators map complex upper half plane to the Siegel upper half space. We also discuss about…

We construct a class of matrix-valued Schr\"odinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods…

We consider fractional Schr\"odinger operators $H=(-\Delta)^\alpha+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha>1$. We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for…

We prove that the wave operators for $n \times n$ matrix Schr\"odinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying…