Reviews in Mathematical Physics

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban’s approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes–Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images.

In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a $\sigma$-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.

In this paper, we study the spectral and scattering theory on the Hamiltonians which govern one-body quantum systems in crossed constant electric and magnetic fields. As one of the spectral features of the Hamiltonians, we show the absence of bound states of theirs under some appropriate conditions on the potentials. Moreover, by using the limiting absorption principle for the Hamiltonians, we give a simplified proof of the asymptotic completeness in short-range scattering.

In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We prove that if the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz-type estimate supported on a compact region in space around the trapped surface, then the components of the Maxwell fields decay uniformly in the entire exterior of the Schwarzschild black hole, including the event horizon. This is shown by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. The proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to decouple the middle components for the Maxwell fields, and would be in particular useful for the non-abelian case of the Yang–Mills equations where the decoupling of the middle components cannot occur. In fact, the estimates for the hereby argument are still valid for the Yang–Mills fields except for the Lie derivatives of the fields that are involved in the proof.