Related papers: Sharp Gaussian upper bounds for Schr\"odinger heat…

We derive a sharp bound on the location of non-positive eigenvalues of Schroedinger operators on the halfline with complex-valued potentials.

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 a unified and optimized proof of the sharp bounds for the Jacobi heat kernel, which were obtained gradually in several papers in recent years. We lay particular emphasis on tracing and estimating all constants appearing throughout…

In this paper, we will prove a gap theorem for four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying…

In the sub-Riemannian manifolds, on the one hand, following Baudoin-Garofalo \cite{BaudoinGarofalo}, the upper bound for heat kernels associated to a class of locally subelliptic operators are given under the generalized curvature-dimension…

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $\kappa$-solutions of the Ricci…

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are,…

We prove a general theorem which provides a strict lower bound on high-temperature Green-Kubo diffusion constants in locally interacting quantum lattice systems, under the assumption of existence of a quadratically extensive almost…

We prove an optimal 4G Theorem for the Gaussian kernel. We also propose a new general method of estimating Schroedinger perturbations of transition densities, and give applications to the Gaussian kernel.

We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative…

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space.…

In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the…

The heat operator with a pure soliton potential is considered and its Green's function, depending on a complex spectral parameter k, is derived. Its boundedness properties in all variables and its singularities in the spectral parameter k…

We study the $L^1\to L^\infty$-decay estimates for dispersive equations in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel…

In this note, we compute the Hadamard coefficients of (algebraically) integrable Schrodinger operators in two dimensions. These operators first appeared in [BL] and [B] in connection with Huygens' principle, and our result completes, in a…

We prove first that the realization $A_{\min}$ of $A:=\mathrm{div}(Q\nabla)-V$ in $L^2(\mathbb{R}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2(\mathbb{R}^d)$ which coincides on…

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set…

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only…

In this work, we study the heat equation with Grushin's operator. We present an expression for its heat kernel, prove its decay in $L^p$ spaces, and that it is an approximation of the identity. As a consequence, the heat semigroup…

The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a…