Related papers: Note on gradient estimate of heat kernel for Schr\…

We prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials.…

Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable…

Let (P(t)) be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the generator of a C_0-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and (W_H(t)) is an…

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…

We study integral kernels of strongly continuous semigroups on Lebesgue spaces over metric measure spaces. Based on semigroup smoothing properties and abstract Morrey-type inequalities, we give sufficient conditions for H\"older or…

We investigate dispersive estimates for the Schr\"odinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy…

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+…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

In this paper, we first prove that the kernel of convolution operator, corresponding the composition of pseudo-differential operator and evolution system associated with the symbol depending on time, satisfies the H\"ormander's condition.…

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or H\"older continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

This paper studies the heat content} for Schr\"odinger operators of the fractional Laplacian $(-\Delta)^{\alpha/2}$, $0<\alpha\leq 2$ in $R^d$, $d\geq 1$. Employing probabilistic and analytic techniques, a small time asymptotic expansion…

Let $\mathcal{L}=-\Delta+V$ be a Schr\"{o}dinger operator, where the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat…

We prove an upper and a lower bound on the rank of the spectral projections of the Schr\"odinger operator $-\Delta + V$ in terms of the volume of the sublevel sets of an effective potential $\frac{1}{u}$. Here, $u$ is the `landscape…

In this paper we consider uncertainty principles for solutions of certain PDEs on H-type groups. We first prove that, contrary to the euclidean setting, the heat kernel on H-type groups is not characterized as the only solution of the heat…

Following the method of Froese and Herbst, we show for a class of potentials V that an eigenfunction $\psi$ with eigenvalue E of the multi-dimensional discrete Schr\"odinger operator H = $\Delta$ + V on \mathbb{Z}^d decays sub-exponentially…

We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvlaues in terms of the $L^q$ norm of $V$.…

We extend a result of Davies and Nath on the location of eigenvalues of Schr\"odinger operators with slowly decaying complex-valued potentials to higher dimensions. In this context, we also discuss various examples related to the…

We study the geometry associated to the Grusin operator G=\Delta_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics…

We compute the Gabor matrix for Schr\"odinger-type evolution operators. Precisely, we analyze the Heat Equation, already presented in \cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix which leads to better…