Letters in Mathematical Physics最新文献

This article establishes a proof of dynamical localization for a random scattering zipper model. The scattering zipper operator is the product of two unitary by blocks operators, multiplicatively perturbed on the left and right by random unitary phases. One of the operator is shifted so that this configuration produces a random 5-diagonal by blocks unitary operator. To prove the dynamical localization for this operator, we use the fractional moments method. We first prove the continuity and strict positivity of the Lyapunov exponents in an annulus around the unit circle, which leads to the exponential decay of a power of the norm of the products of transfer matrices. We then establish an explicit formula of the coefficients of the finite resolvent in terms of the coefficients of the transfer matrices using Schur’s complement. From this, we deduce, through two reduction results, the exponential decay of the resolvent, from which we get the dynamical localization.

We introduce a notion of Lorentzian metric space which drops the boundedness condition from our previous work and argue that the properties defining our spaces are minimal. In fact, they are defined by three conditions given by (a) the reverse triangle inequality for chronologically related events, (b) Lorentzian distance continuity and relative compactness of chronological diamonds, and (c) a distinguishing condition via the Lorentzian distance function. By adding a countably generating condition, we confirm the validity of desirable properties for our spaces including the Polish property. The definition of (pre)length space given in our previous work on the bounded case is generalized to this setting. We also define a notion of Gromov–Hausdorff convergence for Lorentzian metric spaces and prove that (pre)length spaces are GH-stable. It is also shown that our (sequenced) Lorentzian metric spaces bring a natural quasi-uniformity (resp. quasi-metric). Finally, an explicit comparison with other recent constructions based on our previous work on bounded Lorentzian metric spaces is presented.

We show that the n-point, genus-g correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like ((2g - 2 + n)!) as (g rightarrow infty ), which is the expected growth rate. This provides, in particular, an upper bound for many curve counting problems in large genus and serves as a preliminary step for a resurgence analysis.

In this work, we prove rigidity results for complete totally trapped spacelike submanifolds immersed in generalized Robertson–Walker spacetimes. In particular, we obtain uniqueness and non-existence results for totally trapped submanifolds. We use a maximum principle for the (infty )-Laplacian in order to get our results. We also present examples of totally trapped submanifolds in the Schwarzschild black hole spacetime and a surface which is trapped but it is not totally trapped in the product spacetime (-{mathbb {R}}times {mathbb {R}}times {mathbb {H}}^2).

We study the evolution of the roots of a polynomial of degree N, when the polynomial itself is evolving according to the heat flow. We propose a general conjecture for the large-N limit of this evolution. Specifically, we propose (1) that the log potential of the limiting root distribution should evolve according to a certain first-order, nonlinear PDE, and (2) that the limiting root distribution at a general time should be the push-forward of the initial distribution under a certain explicit transport map. These results should hold for sufficiently small times, that is, until singularities begin to form. We offer three lines of reasoning in support of our conjecture. First, from a random matrix perspective, the conjecture is supported by a deformation theorem for the second moment of the characteristic polynomial of certain random matrix models. Second, from a dynamical systems perspective, the conjecture is supported by the computation of the second derivative of the roots with respect to time, which is formally small before singularities form. Third, from a PDE perspective, the conjecture is supported by the exact PDE satisfied by the log potential of the empirical root distribution of the polynomial, which formally converges to the desired PDE as (Nrightarrow infty ). We also present a “multiplicative” version of the the conjecture, supported by similar arguments. Finally, we verify rigorously that the conjectures hold at the level of the holomorphic moments.

It is known in scattering theory that the minimal velocity bound plays a conclusive role in proving the asymptotic completeness of the wave operators. In this study, we prove the minimal velocity bound and other important estimates for the Schrödinger-type operator with fractional powers. We assume that the pairwise potential functions belong to broad classes that include long-range decay and Coulomb-type local singularities.

The Cauchy problem for the massive Dirac equation is studied in the Reissner–Nordström geometry in horizon-penetrating Eddington–Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ordinary differential equations which arise in the separation of variables. Our integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon.

The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associated coarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state (phi ) on an (abstract) spin system with an infinite collection of sites X, we define a universal coarse structure (mathcal {E}_{phi }) on the set X with the property that a state has decay of correlations with respect to a coarse structure (mathcal {E}) on X if and only if (mathcal {E}_{phi }subseteq mathcal {E}). We show that under mild assumptions, the coarsely connected completion ((mathcal {E}_{phi })_{con}) is stable under quasi-local perturbations of the state (phi ). We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit (alpha ) and the coarse structure of the state (psi circ alpha ) where (psi ) is any product state.

The creation of electrically charged states and the resulting electromagnetic fields are considered in spacetime regions in which such experiments can actually be carried out, namely in future-directed light cones. Under the simplifying assumption of external charges, charged states are formed from neutral pairs of opposite charges, with one charge being shifted to light-like infinity. It thereby escapes observation. Despite the fact that this charge moves asymptotically at the speed of light, the resulting electromagnetic field has a well-defined energy operator that is bounded from below. Moreover, due to the spatiotemporal restrictions, the transverse electromagnetic field (the radiation) has no infrared singularities in the light cone. They are quenched and the observed radiation can be described by states in the Fock space of photons. The longitudinal field between the charges (giving rise to Gauss’s law) disappears for inertial observers in an instant. This is consistent with the fact that the underlying longitudinal photons do not manifest themselves as genuine particles. The results show that the restrictions of operations and observations to light cones, which are dictated by the arrow of time, amount to a Lorentz-invariant infrared cutoff.

Using the 3D mirror symmetry we construct a system of polynomials (textsf{T}_s(z)) with integral coefficients which solve the quantum differential equitation of (X=T^{*}operatorname {Gr}(k,n)) modulo (p^s), where p is a prime number. We show that the sequence (textsf{T}_s(z)) converges in the p-adic norm to the Okounkov’s vertex function of X as (srightarrow infty ). We prove that (textsf{T}_s(z)) satisfy Dwork-type congruences which lead to a new infinite product presentation of the vertex function modulo (p^s).