Majorization, 4G Theorem and Schrödinger perturbations

Abstract

Abstract. Schrödinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance, this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations, it is convenient to use an auxiliary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the 1/2-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrödinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.

Non-symmetric Lévy-type operators

“…We refer the reader to [21,Lemma 2.3] for other equivalent descriptions of (3), which involve relations between h and K. We consider J : R d → [0, ∞] such that for some constant c J ∈ [1, ∞) and all x ∈ R d ,…”

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

“…One advantage of that step is that we can define S(t) by an explicit formula that depends on the coefficients of the operator H. Another advantage is that for each t the operator S(t) is a linear bounded operator, which allows us to define analytic functions of argument S(t) via power series (see examples [40,39,37,41,36]) to obtain a semigroup e −itH t≥0 that solves Schrödinger equation with Hamiltonian H. This idea was introduced in [40] where we defined R(t) = exp − i(S(t) − I) and proved that e −itH = lim n→∞ R(t/n) n . Members of O.G.Smolyanov's group employed Chernoff's theorem using integral operators as Chernoff functions to find solutions to parabolic equations in many cases during the last 15 years: see the pioneering papers [46,45], overview [44], several examples [43,11,38,16,47,42,12] to see the diversity of applications, and recent papers [39,13,41,9,5,29,8,36,27,22,24,48,21,7,35]. The solutions obtained were represented in the form of a Feynman formula, i.e.…”

Kato Classes for Lévy Processes

“…In view of the equivalence of (1) and (5) for every Lévy process (see Proposition 3.4 for other description of (1) true for Hunt processes) these conditions should be compared with (2) by its alternative form provided by Proposition 3.6 in a generality of a Hunt process, i.e.,…”

“…On the other hand, if −q 0 is time-independent and the above inequality holds for some N > 0 on the level of densities, then necessarily q ∈ L ∞ (R) (see [5,Corollary 3.4]). Nevertheless, perturbation techniques yield an upper bound by means of an auxiliary density for (unbounded) q from the Kato class if an appropriate 4G inequality for the transition density of the subordinator holds (see [5,Proposition 2.4]). Generators of subordinators generalize fractional derivative operators that are used in statistical physics to model anomalous subdiffusive dynamics (see [16]).…”