Related papers: Relativistic stable operators with critical potent…

We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the…

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a…

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric…

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-\Delta^D_\Omega$ in locally twisted three-dimensional tubes $\Omega$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as…

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…

We introduce a covariant canonical quantization for a particle in curved spacetime that tracks operator-ordering ambiguities. Parameterizing spatial and temporal ordering, we derive a Hermitian Hamiltonian with leading quantum-relativistic…

Let us consider a time-dependent differential operator quadratic with respect to the phase variables. Let us consider a multiplication operator defined with the help of a "small" matrix-valued function. Under suitable conditions, we give an…

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the…

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the $L^p-L^q$ estimates of the associated potential operator obtained recently by Bongioanni and Torrea are…

In this paper we establish the existence and uniqueness of heat kernels to a large class of time-inhomogenous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates,…

We prove some estimations of the correlation of two local observables in quantum spin systems (with Schr\"odinger equations) at large temperature. For that, we describe the heat kernel of the Hamiltonian for a finite subset of the lattice,…

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates…

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of well known two-sided Gaussian heat kernel…

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local…

We construct the fundamental solution (the heat kernel) $p^{\kappa}$ to the equation $\partial_t=\mathcal{L}^{\kappa}$, where under certain assumptions the operator $\mathcal{L}^{\kappa}$ takes one of the following forms, \begin{align*}…

When studying non-symmetric nonlocal operators $$ {\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{\kappa (x, z)}{|z|^{d+\alpha}} d z , $$ where $0<\alpha<2$ and $\kappa (x, z)$ is a…

We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schr\"odinger operator with a non-negative and locally bounded potential $V$ such that $V(x) \to \infty$ as $|x| \to \infty$.…

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation,…

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal…

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…