L estimates for fractional Schrödinger operators with Kato class potentials

Gaussian estimates with best constants for higher-order Schrödinger operators with Kato potentials

“…The sharpness of these estimates depends of course on using the right distance function which is not the Euclidean but, rather, a Finsler distance induced by the operator. The sharp constant σ m , also obtained in [18], was first identified by Evgrafov and Postnikov [17] who obtained short time asymptotics of K 0 (t, x, y) for operators with constant coefficients in R n and so-called strongly convex principal symbol (see definition below).…”

“…Under the assumption that H 0 has constant coefficients they prove that estimate (1) is also valid for the heat kernel K(t, x, y) of H 0 + V . In the very recent article [18] the authors consider the operator (−∆) m + V for Kato potentials V and apply the methods of [15] together with Davies' exponential perturbation technique as adpted in [7] in order to obtain estimates such as (1) for K(t, x, y) with the sharp constant c 2 in the Gaussian exponent.…”

On complex-time heat kernels of fractional Schrödinger operators via Phragmén-Lindelöf principle

“…The power α −1 reflects the scaling relation between time and space in e −zHα and is dictated by the order of the principal symbol of (−∆) α 2 +min{V, 0}. For t > 0 this is seen in (1.3) for V = 0, in (3.1) for α ∈ (0, 2) and V ≥ 0, in (3.15) for V = a|x| −α with a ≥ a * , and in Huang et al [32,Theorem 1.3] when V ∈ K α (R d ) is a perturbation. For complex times the power is expected to be α −1 , too.…”

Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes

“…Recall that, if V is in the higher order Kato class K 2m (R n ) [see (2.4) below for its definition], then V is a Kato type perturbation of P(D) (see [18,66]). Further developments on the local estimates of the type (1.9) can be fund in [4,5,27,38].…”