Letters in Mathematical Physics最新文献

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).

Inspired by the recent paper of Baltazar and Queiroz (J Geom Anal 34:158, 2024. https://doi.org/10.1007/s12220-024-01603-y), in this article, we prove the rigidity for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under a pinching condition, which is independent on the potential function. As a special case of gradient Einstein-type manifold, we also give a rigidity result of ((m,rho ))-quasi-Einstein manifold with boundary.

We investigate the count of meromorphic differentials on the Riemann sphere possessing a single zero, multiple poles with prescribed orders and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties for the solution formula as well as related intersection numbers.

We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application, we reprove and extend some theorems about eigenvalue asymptotics of Schrödinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.

The Berezin–Karpelevich integral is a double integral over unitary matrices which plays the role of the Itzykson–Zuber integral in rectangular matrix models. We obtain a topological expansion of the Berezin–Karpelevich integral in terms of monotone Hurwitz numbers and obtain from this certain combinatorial identities.