Related papers: On Dunkl Schr\"odinger semigroups with Green bound…

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x, $$ let…

In this paper, we study the Schr\"odinger operator $\Delta-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2\beta)}$, $b, \beta>0$. We obtain…

On the Euclidean space $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k\geq 0$, and the associated measure $dw(\mathbf x)=\prod_{\alpha\in R} |\langle \mathbf x,\alpha\rangle|^{k(\alpha)}d\mathbf x$ we…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. We denote by $dw(\mathbf{x})=\Pi_{\alpha \in R}|\langle \mathbf{x},\alpha…

Consider the Schr\"odinger operator $ \mathcal L^V=-\Delta+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$…

Let $m\in\mathbb N$, $P(D):=\sum_{|\alpha|=2m}(-1)^m a_\alpha D^\alpha$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the…

We consider the Schr{\"o}dinger operator H = --$\Delta$ + V (|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their…

Let $\Delta$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-t\Delta}$ generated by $\Delta$.…

We give the upper and the lower estimates of heat kernels for Schr\"odinger operators $H=-\Delta+V$, with nonnegative and locally bounded potentials $V$ in $\mathbb{R}^d$, $d \geq 1$. We observe a factorization: the contribution of the…

We prove that the heat kernel associated to the Schr\"odinger type operator $A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq…

Let $L_k=-\Delta_k+V$ be the Dunk- Schr\"{o}dinger operators, where $\Delta_k=\sum_{j=1}^dT_j^2$ is the Dunkl Laplace operator associated to the dunkl operators $T_j$ on $\mathbb{R}^d$ and $V$ is a nonnegative potential function. In the…

We study the heat kernel $p(x,y,t)$ associated to the real Schr\"odinger operator $H = -\Delta + V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the case that…

Consider the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse H\"older condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B…

We investigate the Dirichlet problem associated to the Schr\"odinger operator $\mathcal L=-\Delta_{\mathbb{H}^n}+V$ on Heisenberg group $\mathbb H^n$: \begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad &{\rm in \,\…

Let $L=-\Delta+V$ be a Schr\"odinger operator, where the potential $V$ belongs to the reverse H\"older class. By the subordinative formula, we introduce the fractional heat semigroup $\{e^{-t{L}^\alpha}\}_{t>0}, \alpha>0$, associated with…

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary and let $K\subset\partial\Omega$ be either a $C^2$ submanifold of the boundary of codimension $k<N$ or a point. In this article we study various problems related to the…

In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ satisfies $$ k(t,x,y) \leq c_1e^{\lambda_0 t+…

A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Denote by $dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x…