Communications in Mathematical Physics最新文献

We consider the spectral decomposition of singularities of integrals and their integrands. Our results apply to any integral of Euler–Mellin type, and thus especially to every scalar Feynman integral. Specifically we provide for both the integrand and integral respectively; two explicit constructions of the characteristic variety and characteristic cycle of the constructible function and D-module they are associated with. From this we also obtain the singular locus or Landau singularities of the integral. En route we give a simple procedure to compute the local Euler obstruction function of a variety, and using this, to compute the Euler characteristic of the complex link of a Whitney stratum.

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian (Ntimes N) non-Hermitian random band matrices with a bandwidth W. Given (W,Nrightarrow infty ), we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at (Wsim sqrt{N}): it coincides with that for Ginibre ensemble for (Wgg sqrt{N}), and is factorized as (1ll Wll sqrt{N}). The result is the first step toward the proof of Anderson’s type transition for non-Hermitian random band matrices.

The first goal of this paper is to give a precise and simple definition for off-shell Bethe vectors in a generic (mathfrak {g})-invariant integrable model for (mathfrak {g}=mathfrak {gl}_n), (mathfrak {o}_{2n+1}), (mathfrak {sp}_{2n}) and (mathfrak {o}_{2n}). We prove from our definition that the off-shell Bethe vectors indeed become on-shell when the Bethe equations are obeyed. Then, we show that some properties for these off-shell Bethe vectors, such as the action formulas of monodromy entries on these vectors, their rectangular recurrence relations and their coproduct formula, are a consequence of our definition.

We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.

Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs. In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-through algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev’s quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing.

We consider classical/quantum correspondence in Lindblad evolution with jump operators for which the corresponding Fokker–Planck equation is subelliptic. This allows us to consider the physical model proposed by Zurek and Paz [15], and to extend some of the recent mathematical results of Hernandez et al. [8, 9], Galkowski and Zworski [5], and Li [11], where the diffusion term in the Fokker–Planck equation was assumed elliptic. We consider the case where the jump operators (ell _j) in the Lindbladian are linear functions of x, and place an assumption which implies that the Hörmander condition holds for the resulting Fokker–Planck equation. By constructing a suitable parametrix for this equation we show that the semiclassical derivative estimates established in [5] and [11] for elliptic diffusion also hold in the subelliptic case, with global bounds in (L^p) for all (1le ple infty ).

We consider a gravitational analogue of the Schwinger effect in a cosmological context. While the Schwinger effect is usually attributed to a static electric background, its derivation is actually based on a switching on/off of the electric field in the infinite past/future. Motivated by this, and our previous work on particle production in a gravitational background, we consider a long pulse of the gravitational field in an FLRW-spacetime, thus simulating a static background. We rigorously derive particle production by a novel application of the Heun equation. In fact, the recently obtained connecting formulas between its local solutions can be used to determine the Bogolyubov coefficients, and subsequently the particle production probabilities, in the limit of an infinitely long pulse. The particle production in the FLRW-model is found to have a lower threshold on the late time frequencies, which can be related to the duration of the time interval of the switching on/off the background field. For large frequencies and in the spatially flat case, we find a Planckian frequency spectrum whose temperature is inversely proportional to the duration of scale change. We compare our findings to Schwinger’s result on particle production for a long pulse of an electromagnetic background field, for which we also include a detailed derivation.

The compressible Euler–Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy solutions of the multidimensional Euler–Riesz equations with large initial data of spherical symmetry. We consider both attractive and repulsive interactions for a wide range of ( {Riesz}) and ( {logarithmic}) potentials for dimensions larger than or equal to two. This is achieved by the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier–Stokes–Riesz equations. The strong convergence of the vanishing viscosity solutions is achieved through delicate uniform estimates in (L^p). It is observed that, even if the attractive potential is super-Coulomb, no concentration is formed near the origin in the inviscid limit. Moreover, we prove that the nonlinear stability of global finite-energy solutions for the Euler–Riesz equations is ( {unconditional}) under a spherically symmetric perturbation around the steady solutions. Unlike the Coulomb case where the potential can be represented locally, the singularity and regularity of the nonlocal radial Riesz potential near the origin require careful analysis, which is a crucial step. We also establish uniform energy estimates of global solutions for not only the Riesz potential but also the limiting end-point case: the logarithmic potential. Finally, unlike the Coulomb case, a Grönwall type estimate is required to overcome the difficulty of the appearance of boundary terms in the sub-Coulomb case and the singularity of the super-Coulomb potential. Furthermore, we prove the nonlinear stability of global finite-energy solutions for the compressible Euler–Riesz equations around steady states by employing concentration compactness arguments. Steady states properties are obtained by variational arguments connecting to recent advances in aggregation-diffusion equations.

Given a (hbox {II}_1) factor M, a (hbox {W}^*)-subalgebra (Qsubset M) is compressible if for any (varepsilon >0) there exists a finite set of unitary elements ({mathcal {U}}_0subset {mathcal {U}}(M)=mathcal U(Motimes 1)) such that (Vert frac{1}{|{mathcal {U}}_0|}sum _{uin mathcal U_0} uxu^* -E_{1otimes {mathbb {M}}_K({mathbb {C}})}(x)Vert le varepsilon ), (forall Kge 1), (forall xin (Qotimes {mathbb {M}}_K({mathbb {C}}))_1). Any (hbox {W}^*)-subalgebra Q in a (hbox {II}_1) factor M which admits a diffuse (hbox {W}^*)-algebra (Q_0subset M) that’s free independent to Q, is compressible in M. We prove that if (Qsubset M) is compressible, then (_NL^2M_Q) contains a copy of the coarse (N-Q) bimodule for any AFD subalgebra (Nsubset M). We use this result to provide examples of inclusions of factors (Msubset {mathcal {M}}) that are ergodic but not AFD-ergodic, even after stabilizing by (mathcal B(ell ^2{mathbb {N}})).

This paper investigates exponential mixing of the invariant measure for randomly forced nonlinear Schrödinger equation, with damping and random noise localized in space. Our study emphasizes the crucial role of exponential asymptotic compactness and control properties in establishing the ergodic properties of random dynamical systems. This work extends the series [16, 46] on the statistical behavior of randomly forced dispersive equations.