[Paper Review] A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs

Batu Güneysu, Matthias Keller+1 more|arXiv (Cornell University)|Jan 7, 2013

Spectral Theory in Mathematical Physics36 references9 citations

TL;DR

This paper establishes a Feynman-Kac-Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs by introducing a general framework for magnetic potentials and stochastic line integrals. The key contribution is a probabilistic representation of the semigroup associated with these operators, enabling new proofs of Kato's inequality, Golden-Thompson inequality, and explicit quadratic form domain characterizations.

ABSTRACT

In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödinger 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.

Motivation & Objective

• To develop a general and natural framework for defining magnetic Schrödinger operators on arbitrary weighted graphs.
• To introduce a consistent notion of stochastic line integrals with respect to magnetic potentials on graphs.
• To establish a Feynman-Kac-Itô formula linking discrete magnetic operators with stochastic processes on graphs.
• To apply the formula to derive new analytical results, including Kato's inequality and Golden-Thompson inequality.
• To characterize the quadratic form domains for a broad class of potentials on graphs.

Proposed method

• Defining a general class of potentials that allow self-adjoint realization of magnetic Schrödinger operators on weighted graphs.
• Introducing a notion of stochastic line integrals along random paths on graphs using magnetic edge functions.
• Constructing the Feynman-Kac-Itô formula via the semigroup representation involving the stochastic exponential of the line integral.
• Using Mosco convergence to justify approximation results for quadratic forms and semigroups.
• Applying Pitt's theorem and abstract Golden-Thompson inequalities to derive domination and trace estimates.
• Establishing the equivalence of the maximal and form-closed quadratic forms under mild potential conditions.

Experimental results

Research questions

• RQ1 Can a Feynman-Kac-Itô formula be rigorously formulated for magnetic Schrödinger operators on discrete graphs?
• RQ2 What is a general and consistent definition of stochastic line integrals for magnetic potentials on graphs?
• RQ3 Under what conditions is the magnetic Schrödinger operator on a graph essentially self-adjoint?
• RQ4 How can the quadratic form domain of such operators be explicitly characterized?
• RQ5 What probabilistic and analytical inequalities can be derived from the Feynman-Kac-Itô formula on graphs?

Key findings

• The Feynman-Kac-Itô formula is established for magnetic Schrödinger operators on arbitrary weighted graphs, linking the semigroup to stochastic processes with magnetic line integrals.
• A general class of potentials is identified under which the magnetic Schrödinger operator is well-defined as a self-adjoint operator.
• Kato's inequality is proven as a consequence of the Feynman-Kac-Itô formula, showing domination of the magnetic semigroup over the non-magnetic one.
• The Golden-Thompson inequality is derived in an abstract form, providing a trace estimate for the semigroup of the sum of two operators.
• The quadratic form domain of the magnetic Schrödinger operator is explicitly characterized as the maximal form domain when the potential is bounded below.
• The maximal and form-closed quadratic forms coincide for potentials bounded below, ensuring a consistent operator realization.