Conference Agenda

We consider the Schrödinger operator of the form $H=-\Delta+V$ acting in $L^2(R^d,dx)$, $d \geq 1$, where the potential $V:R^d \to [0,\infty)$ is a locally bounded function. The corresponding Schrödinger semigroup $\big\{e^{-tH}: t \geq 0\big\}$ consists of integral operators, i.e. $$ e^{-tH} f(x) = \int_{R^d} u_t(x,y) f(y) dy, \quad f \in L^2(R^d,dx), \ t>0. $$

I will present new estimates for heat kernel of $u_t(x,y)$. Our results show the contribution of the potential is described separately for each spatial variable, and the interplay between the spatial variables is seen only through the Gaussian kernel.

This estimates will be presented on two common classes of potentials. For confining potentials we get two sided estimates and for decaying potentials we get new upper estimate.

Methods we used to estimate kernel of semigroup allow to easily obtain sharp estimates of ground state for slowly varying potentials.

The talk is based on joint work with Kamil Kaleta [BK] and my work [B].

[BK] M. Baraniewicz, K. Kaleta, Integral kernels of Schrödinger semigroups with nonnegative locally bounded potentials, Studia Mathematica 275, 2024

[B] M. Baraniewicz, Estmates of ground state for classical Schrödinger operator. To appear, 2024+.

We present results of our investigation of a particular discrete-time counterpart of the Feynman--Kac semigroup with a confining potential in a countably infinite space. We focus on Markov chains with the direct step property, which is satisfied by a wide range of typically considered kernels. In our joint work with Wojciech Cygan, Ren\'e Schilling and Kamil Kaleta, we introduce the concept of progressive intrinsic ultracontractivity (pIUC) and investigate links between pIUC of Feynman--Kac semigroups, their uniform quasi-ergodicity and uniform ergodicity of their intrinsic semigroups. In particular, we study certain discrete analogues of examples common in literature and estimate the rates of convergence of these properties.

The following Schrödinger operator $$ K = K_0 + V, $$ where $$ K_0 = \sqrt{-\frac{\partial^2}{\partial x_1^2}} + \sqrt{-\frac{\partial^2}{\partial x_2^2}} $$ is an example of a nonlocal, anisotropic, singular Lévy operator. We consider potentials $V : \mathbb{R}^2 \to \mathbb{R}$ such that $V(x)$ goes to infinity as $|x| \to \infty$. The operator $-K_0$ is a generator of a process $X_t = (X_t^{(1)}, X_t^{(2)})$, sometimes called cylindrical, such that $X_1^{(1)}$, $X_2^{(2)}$ are independent symmetric Cauchy processes in $\mathbb{R}$.

We define the Feynman-Kac semigroup $$ T_t f(x) = E^x \left( \exp \left( -\int_0^t V(X_s) \, ds \right) f(X_t) \right). $$ Operators $T_t$ are compact for every $t > 0$. There exists an orthonormal basis $\{ \phi_n \}_{n=1}^{\infty}$ in $L^2 (\mathbb{R}^2)$ and a corresponding sequence of eigenvalues $\{\lambda_n \}_{n=1}^{\infty}$, $0<\lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \dots$, $\lim_{n \to \infty} \lambda_n = \infty$ such that $T_t \phi_n = e^{-\lambda_n t} \phi_n$. We can assume that $\phi_1$ is positive and continuous on $\mathbb{R}^2$. The main result I would like to present concerns estimates for $\phi_1$ and intrinsic ultracontractivity of the semigroup $T_t$ under certain conditions on the potential $V$.

On a smooth (possibly) non-compact geodesically complete connected Riemannian manifold, we investigate a Harnack inequality for the time-independent Schrödinger operator provided its potential is in the Kato class and Ricci curvature is bounded from below by a function in the Kato class. The proof is based on probabilistic Li-Yau type inequalities established for corresponding diffusion semigroup by martingale methods.