A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

Abstract

In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödi-nger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials. Mathematics Subject Classification

Magnetic-Sparseness and Schrödinger Operators on Graphs

“…Again we write K θ α := K 2,θ α and K θ 0 + := K 2,θ 0 + . In [GKS,Proposition 2.8] it is shown that in the case p = 2 the forms Q (comp) 2,b,θ,q,m are closable in ℓ 2 (X, m) for any q ∈ K θ α with α ∈ (0, 1) and we denote the closure by Q θ = Q 2,b,θ,q,m . Furthermore,…”

“…Note that, in the case q = 0, even if the value of Q θ (f ) does not depend on m for f ∈ C c (X), its domain D(Q θ ) does depend on m. Moreover, by [GKS,Theorem 2.12]…”

A Rigorous Mathematical Construction of Feynman Path Integrals for the Schrödinger Equation with Magnetic Field

“…Let us also mention that the study of heat semigroup with magnetic field via Feynman-Kac-Itô formula [59,61] has been extended to the case of fractals, e.g. [33], graphs [30], and manifolds [11,29,60,14].…”

Compressed Resolvents and Reduction of Spectral Problems on Star Graphs

“…Since it seems impossible to list all relevant literature, we mention the three monographs [5,26,33], whose bibliographies reflect the intense research activities in this area very well. The vast majority of these works considers concrete operators on graphs such as Sturm-Liouville and Schrödinger operators [2,3,16,25,37], Laplace and related operators [29], Krein strings [13], Stieltjes strings [30,31], canonical systems and Dirac operators [1,8]. More abstract methods were developed in [32,35].…”