Reviews in Mathematical Physics

For a subquadratic symbol H on ℝ^d × ℝ^d = T*(ℝ^d), the quantum propagator of the time dependent Schrödinger equation is a Semiclassical Fourier-Integral Operator when Ĥ = H(x, ℏD_x) (ℏ-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when ℏ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by H, with a full uniform asymptotic expansion in ℏ for the amplitude.

In this paper, our goal is to revisit this well-known and fundamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global L² estimates when ℏ is small and time t is large. One of the simplest choice of the phase is known in chemical physics as Herman–Kluk formula. Moreover, we prove that the semiclassical expansion for the propagator is valid for where δ > 0 is a stability parameter for the classical system.

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Generalized Pauli–Fierz Hamiltonian with Kato-class potential K_PF in nonrelativistic quantum electrodynamics is defined and studied by a path measure. K_PF is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.

Let be a formally self-adjoint (elliptic) operator in L² (ℝⁿ), n ≥ 2. The real coefficients a_{j, k}(x) = a_{k, j}(x) are assumed to be bounded and to coincide with -Δ outside of a ball. The paper deals with two topics: (i) An eigenfunction expansion theorem, proving in particular that H is unitarily equivalent to -Δ, and (ii) Global spacetime estimates for the associated inhomogeneous wave equation, proved under suitable ("nontrapping") additional assumptions on the coefficients. The main tool used here is a Limiting Absorption Principle (LAP) in the framework of weighted Sobolev spaces, which holds also at the threshold.