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Jacky Cresson, Laurent Nottale, Thierry Lehner
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0037265

Abstract:
Using the formalism of stochastic embedding developed by Cresson and Darses [J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two definitions of a stochastic Newton equation called differential and variational. We first prove a stochastic virial theorem that is a natural generalization of the classical case. The stochasticity modifies the virial relation by adding a potential term called the induced potential, which corresponds in quantum mechanics to the Bohm potential. Moreover, the differential stochastic Newton equation naturally provides an action functional that satisfies a stochastic Hamilton–Jacobi equation. The real part of this equation corresponds to the classical Hamilton–Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic process solutions of the stochastic Newton equation. It is proved that this density satisfies a nonlinear Schrödinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad hoc “dark potential” used to recover a flat rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by Da Rocha and Nottale [Chaos, Solitons Fractals, 16(4), 565–595 (2003)] where the emergence of the “dark potential” is seen as a consequence of the fractality of space in the context of the scale relativity theory.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0031564

Abstract:
In this paper, we study Liouville theorems of D-solutions to the stationary magnetohydrodynamic system in a slab. We will prove trivialness of the velocity and the magnetic field with various boundary conditions. In some boundary conditions, the additional assumption that the horizontal angular component(s) of the velocity or (and) the magnetic field is (are) axially symmetric is needed. More precisely, five types of boundary conditions will be considered: the vertical periodic boundary condition for the velocity and the magnetic field, the Navier-slip boundary condition for the velocity, the perfectly conducting or insulating boundary condition for the magnetic field, the non-slip boundary condition for the velocity, and the perfectly conducting or insulating boundary condition for the magnetic field. One of our innovations is that we do not impose finite Dirichlet integral assumption on the magnetic field compared with previous works.
Jung-Chao Ban, Wen-Guei Hu, Song-Sun Lin, Yin-Heng Lin
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0007365

Abstract:
This work introduces constructive and systematic methods for verifying the topological mixing and strong specification (or strong irreducibility) of two-dimensional shifts of finite type. First, we define transition matrices on infinite strips of width n for all n ≥ 2. To determine the primitivity of the transition matrices, we introduce the connecting operators that reduce the high-order transition matrices to lower-order transition matrices. Then, two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. Then, the primitivity, corner-extendability, and crisscross-extendability are used to demonstrate the topological mixing. Finally, we show that the hole-filling condition yields the strong specification property. The application of all the above-mentioned conditions can be verified in a finite number of steps.
, Christopher Shirley
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0033583

Abstract:
Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular, we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0039732

Abstract:
In this work, we introduce the Z3-graded differential algebra, denoted by Ω(GL̃q(2)), treated as the Z3-graded quantum de Rham complex of Z3-graded quantum group GL̃q(2). In this sense, we construct left-covariant differential calculi on the Z3-graded quantum group GL̃q(2).
, Alexander V. Turbiner, Willard Miller
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0050572

Abstract:
As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij≡rij2, the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0041130

Abstract:
Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [Ballesteros et al., Classical Quantum Gravity 25, 165005 (2008)], the Taub–NUT system [Ballesteros et al., SIGMA 7, 048 (2011)], and all the 17 superintegrable systems for the four types of Darboux spaces as determined by Kalnins et al. [J. Math. Phys. 44, 5811–5848 (2003)].
, , E. Presutti
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0049680

Abstract:
We study the stationary measures of Ginzburg–Landau (GL) stochastic processes, which describe the magnetization flux induced by the interaction with reservoirs. To privilege simplicity to generality, we restrict to quadratic Hamiltonians where almost explicit formulas can be derived. We discuss the case where reservoirs are represented by boundary generators (mathematical reservoirs) and compare with more physical reservoirs made by large-infinite systems. We prove the validity of the Fick law away from the boundaries. We also obtain in the context of the GL models a mathematical proof of the Darken effect, which shows uphill diffusion of carbon in a specimen partly doped with the addition of Si.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0015534

Abstract:
This paper deals with (globally) random substitutions on a finite set of prototiles. Using renormalization tools applied to objects from operator algebras, we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic Rd action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schrödinger operators obtained from aperiodic tilings in the construction.
Xuan Liu,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0045390

Abstract:
We consider the fourth-order Schrödinger equation i∂tu + Δ2u + μΔu + λ|u|αu = 0 in HsRN, with N≥1,λ∈C, μ = ±1 or 0, 0 < s < 4, 0 < α, and (N − 2s)α < 8. We establish the local well-posedness result in Hs(RN) by applying Banach’s fixed-point argument in spaces of fractional time and space derivatives. As a by-product, we extend the existing H2 local well-posedness results to the whole range of energy subcritical powers and arbitrary λ∈C.
, Ruiao Hu
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0045010

Abstract:
This paper introduces an energy-preserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics, and the Euler–Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. We also present an Eulerian averaged SFLT model based on decomposing the Eulerian solutions of the energy-conserving SFLT model into sums of their expectations and fluctuations.
, Maria Radosz, Angel Harb, Aaron DeLeon, Alan Baza
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0042690

Abstract:
This paper considers the relativistic motion of charged particles coupled with electromagnetic fields in the higher-order theory proposed by Bopp, Landé–Thomas, and Podolsky. We rigorously derive a world line integral expression for the self-force of the charged particle from a distributional equation for the conservation of four-momentum only. This naturally leads to an equation of motion for charged particles that incorporates a history-dependent self-interaction. We show additionally that the same equation of motion follows from a variational principle for retarded fields. Our work thus gives a rigorous vindication of an expression for the self-force first proposed by Landé and Thomas, studied by Zayats for straight line motion, and, more generally, obtained by Gratus, Perlick, and Tucker on the basis of an averaging axiom.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0053606

Abstract:
We study the Boussinesq hierarchy in the geometric context of the theory of bi-Hamiltonian manifolds. First, we recall how its bi-Hamiltonian structure can be obtained by means of a process called bi-Hamiltonian reduction, choosing a specific symplectic leaf S of one of the two Poisson structures. Then, we introduce the notion of a bi-Hamiltonian S-hierarchy, that is, a bi-Hamiltonian hierarchy that is defined only at the points of the symplectic leaf S, and we show that the Boussinesq hierarchy can be interpreted as the reduction of a bi-Hamiltonian S-hierarchy.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0043743

Abstract:
Susskind–Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind–Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In this paper, the quantization map related to the modified Susskind–Glogower coherent states is exploited, which naturally leads to a particular representation of the su(1,1) Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind–Glogower ones by extending the indices of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind–Glogower-I and Susskind–Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the su(1,1) and su(2) Lie algebras as generators of the SU(1, 1) and SU(2) unitary irreducible representations, respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.
Mi-Ran Choi, Younghoon Kang, Young-Ran Lee
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0053132

Abstract:
We show the global well-posedness of the nonlinear Schrödinger equation with periodically varying coefficients and a small parameter ɛ > 0, which is used in optical-fiber communications. We also prove that the solutions converge to the solution for the Gabitov–Turitsyn or averaged equation as ɛ tends to zero.
Benjamin Dodson, , Thomas Spencer
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0042321

Abstract:
In this paper, we continue our study [B. Dodson, A. Soffer, and T. Spencer, J. Stat. Phys. 180, 910 (2020)] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on R was proved for real analytic data. Here, we prove global well-posedness for the 1D NLS with initial data lying in Lp for any 2 < p < ∞, provided that the initial data are sufficiently smooth. We do not use the complete integrability of the cubic NLS.
Nakao Hayashi,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0052299

Abstract:
We study the asymptotic behavior of solutions to the Cauchy problem for the higher-order anisotropic nonlinear Schrödinger equation in two space dimensions i∂tu+12Δu−14∂x14u=λuu, t > 0, x∈R2, with initial data u0,x=u0x, x∈R2, where λ∈R. We will show the modified scattering for solutions. We continue to develop the factorization techniques, which were started in the papers of N. Hayashi and P. I. Naumkin [Z. Angew. Math. Phys. 59(6), 1002–1028 (2008); J. Math. Phys. 56(9), 093502 (2015)], N. Hayashi and T. Ozawa [Ann. I.H.P.: Phys. Theor. 48, 17–37 (1988)], and T. Ozawa [Commun. Math. Phys. 139(3), 479–493 (1991)]. The crucial point of our approach presented here is the L2-boundedness of the pseudodifferential operators.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0047754

Abstract:
We formulate the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure’s gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution’s gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters, including the angular speed.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0040051

Abstract:
Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain interactions that “glue” two or more regions of a chain. The resulting fold topology is widely believed to be a determinant of biomolecular properties and function. Recently, knot theory has been extended to describe the topology of folded linear chains, such as proteins and nucleic acids. To classify and distinguish chain topologies, algebraic structure of quandles has been adapted and applied. However, the approach is limited as apparently distinct topologies may end up having the same number of colorings. Here, we enhance the resolving power of the quandle coloring approach by introducing Boltzmann weights. We demonstrate that the enhanced coloring invariants can distinguish fold topologies with an improved resolution.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0048929

Abstract:
We study stealth black hole perturbations in shift symmetric kinetic gravity braiding and obtain its analogous Regge–Wheeler and Zerilli master equations for the odd and even parity sectors. We show that the nontrivial hair of static and spherically symmetric stealth black holes contributes only an additional source term to the even parity master equation. Furthermore, we derive exact solutions to the monopolar and dipolar perturbations and show that they are generally pathological non-gauge modes or else reduce to the pure-gauge low-order multipoles of general relativity.
, Yangmin Xiong
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0020351

Abstract:
The aim of this paper is to consider the asymptotic dynamics of solutions to 2D MHD equations when the external forces contain some hereditary characteristics. First, we establish, respectively, the well-posedness of strong solutions and weak solutions; then, the process Ũ(⋅,⋅) generated by the weak solutions is constructed in MH2(=H×LH2); and finally, we analyze the long-time behavior of the weak solutions by proving the existence of a compact pullback attractor.
, Anas A. Rahman
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0039887

Abstract:
We outline a relation between the densities for the β-ensembles with respect to the Jacobi weight (1 − x)a(1 + x)b supported on the interval (−1, 1) and the Cauchy weight (1−ix)η(1+ix)η̄ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for β = 2 and of degree five for β = 1 and 4, analogs of which are already known for the Jacobi weight xa(1 − x)b supported on (0, 1). We concentrate on the case a = b [Jacobi weight on (−1, 1)] and η real (Cauchy weight) since the density is then an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on (−1, 1) and/or the moments of the Cauchy weight. Particular attention is paid to the case β = 2 and the Jacobi weight on (−1, 1) in the symmetric case a = b, which in keeping with a recent result obtained by Assiotis et al. (“Moments of generalised Cauchy random matrices and continuous-Hahn polynomials,” Nonlinearity (to be published), arXiv:2009.04752) for the β = 2 case of the symmetric Cauchy weight (parameter η real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with η = −β(N − 1)/2 − 1 − α, after appropriately scaling α proportional to N, we use differential equations to compute terms in the 1/N2 (1/N) expansion of the resolvent for β = 2 (β = 1, 4).
Bijan Bagchi, Rahul Ghosh
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0043854

Abstract:
We investigate the most general form of the one-dimensional Dirac Hamiltonian HD in the presence of scalar and pseudoscalar potentials. To seek embedding of supersymmetry (SUSY) in it, as an alternative procedure to directly employing the intertwining relations, we construct a quasi-Hamiltonian K, defined as the square of HD, to explore the consequences. We show that the diagonal elements of K under a suitable approximation reflect the presence of a superpotential, thus proving a useful guide in unveiling the role of SUSY. For illustrative purposes, we apply our scheme to the transformed one-dimensional version of the planar electron Hamiltonian under the influence of a magnetic field. We generate spectral solutions for a class of isochronous potentials.
Michael Rushka, M. A. Esrick, W. N. MathewsJr.,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0021013

Abstract:
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.
E. Choreño, R. Valencia,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0027957

Abstract:
In this paper, we study a general Hamiltonian with a linear structure given in terms of two different realizations of the SU(1, 1) group. We diagonalize this Hamiltonian by using the similarity transformations of the SU(1, 1) and SU(2) displacement operators performed to the su(1, 1) Lie algebra generators. Then, we compute the Berry phase of a general time-dependent Hamiltonian with this general SU(1, 1) linear structure.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0054292

Abstract:
Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0048481

Abstract:
In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading 1/N correction to the density. From earlier literature, the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying, in particular, the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterization of the moments and covariances of monomial linear statistics is through recurrence relations. We also extend recent work, which begins with the β-ensembles in the high-temperature limit and constructs a family of tridiagonal matrices referred to as α-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d. (Independent Identically Distributed) gamma distributed random variables. From this, we can supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0037375

Abstract:
The purpose of this article is to prove the existence of solution to a nonlinear nonlocal elliptic problem with a singularity and a discontinuous critical nonlinearity, which is given as (−Δ)psu= μg(x,u)+λuγ+H(u−α)ups*−1inΩ,u>0inΩ, with the zero Dirichlet boundary condition. Here, Ω⊂RN is a bounded domain with Lipschitz boundary, s ∈ (0, 1), 2

0. Under suitable assumptions on the function g, the existence of solution to the problem has been established. Furthermore, it will be shown that as α → 0+, the sequence of solutions of the problem for each such α converges to a solution of the problem for which α = 0.

W.-Q. Tao
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0051641

Abstract:
For the centrally extended Heisenberg double of SL2, its center is determined, the central factor algebras are described, and classifications of simple Harish-Chandra modules, simple Whittaker modules, and simple quasi-Whittaker modules are obtained. Two classes of simple weight modules with infinite-dimensional weight spaces are given. We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional sl2-modules with finite multiplicities.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0040819

Abstract:
We study non-Markovian stochastic differential equations with additive noise characterized by a Poisson point process with arbitrary pulse shapes and exponentially distributed intensities. Specifically, analytic results concerning transitions between different correlation regimes and the long-time asymptotic probability distribution functions are shown to be controlled by the shape of the pulses and dissipative parameter as time progresses. This program is motivated by the study of stochastic partial differential equations perturbed by space Poisson disorder and becomes the main focus of applications of the present exact functional approach.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0047496

Abstract:
In quantum estimation theory, the Holevo bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Holevo bound is defined by a solution of a minimization problem, and in general, an explicit solution is not known. When the dimension of Hilbert space is 2 and the number of parameters is 2, an explicit form of the Holevo bound was given by Suzuki. In this paper, we focus on a logarithmic derivative that lies between the symmetric logarithmic derivative (SLD) and the right logarithmic derivative parameterized by β ∈ [0, 1] to obtain lower bounds of the weighted trace of covariance of an unbiased estimator. We introduce the maximum logarithmic derivative bound as the maximum of bounds with respect to β. We show that all monotone metrics induce lower bounds, and the maximum logarithmic derivative bound is the largest bound among them. We show that the maximum logarithmic derivative bound has explicit solution when the d dimensional model has d + 1 dimensional D invariant extension of the SLD tangent space. Furthermore, when d = 2, we show that the maximization problem to define the maximum logarithmic derivative bound is the Lagrangian duality of the minimization problem to define the Holevo bound and is the same as the Holevo bound. This explicit solution is a generalization of the solution for a two-dimensional Hilbert space given by Suzuki. We also give examples of families of quantum states to which our theory can be applied not only for two-dimensional Hilbert spaces.
Na Wang, Linjie Shi
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0033847

Abstract:
In this paper, we first calculate the orthogonal basis of the vector space spanned by eiejeke0|0⟩, where ej are the generators of the affine Yangian of gl(1). The elements of this orthogonal basis correspond to three dimensional bosons. Then, we calculate the Schur functions of plane partitions of 4, we find that the plane partitions become Young diagrams, and the Schur functions on plane partitions become Schur functions on Young diagrams when h1 = 1, h2 = −1, and h3 = 0.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0038838

Abstract:
Several techniques of generating random quantum channels, which act on the set of d-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show that they are mathematically equivalent. We discuss under which conditions they give the uniform Lebesgue measure on the convex set of quantum operations and compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute the mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity, and the 2-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed, and their spectral properties are studied using the Bloch representation of a classical probability vector.
Chengliang Tan,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0040204

Abstract:
In this paper, we describe the singularity patterns of d-PIV in detail and calculate the algebraic entropies of d-PIV and d-PV, which are shown to be both zeros. However, for the discrete equations that are similar with d-PIV, not all algebraic entropies are zeros. An example shows that even if the equation passes the singularity confinement test, it still has a positive algebraic entropy. We also explore the influence of cyclic patterns and anti-confined patterns in the calculation.
Kui Zhang,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0023227

Abstract:
In this paper, we develop asymptotic theory for the mixing detection methodology proposed by Magdziarz and Weron [Phys. Rev. E 84, 051138 (2011)]. The assumptions cover a broad family of Gaussian stochastic processes, including fractional Gaussian noise and the fractional Ornstein–Uhlenbeck process. We show that the asymptotic distribution and convergence rates of the detection statistic may be, respectively, Gaussian or non-Gaussian and standard or nonstandard depending on the diffusion exponent. The results pave the way for mixing detection based on a single observed sample path and by means of robust hypothesis testing.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0049355

Abstract:
The infinite affine Lie algebras of type ABCD, also called gl̂(∞), ô(∞), and sp̂(∞), are equivalent to subalgebras of the quantum W1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (Â∞), a Majorana fermion (B̂∞ and D̂∞), or a symplectic boson (Ĉ∞). Explicit formulas for the action of the quantum W1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B̂∞, a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.
Ya-Hong Guo, , Na Cui
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0041372

Abstract:
In this paper, we study the following critical fractional Schrödinger equations with the magnetic field: ε2s(−Δ)A/εsu+V(x)u=λf(|u|)u+|u|2s*−2uinRN, where ɛ and λ are positive parameters and V:RN→R and A:RN→RN are continuous electric and magnetic potentials, respectively. Under a global assumption on the potential V, by applying the method of Nehari manifold, Ekeland’s variational principle, and Ljusternick–Schnirelmann theory, we show the existence of ground state solution and multiplicity of non-negative solutions for the above equation for all sufficiently large λ and small ɛ. In this problem, f is only continuous, which allows us to study larger classes of nonlinearities.
V. Prokofev,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0051713

Abstract:
We consider solutions of the matrix Kadomtsev–Petviashvili (KP) hierarchy that are elliptic functions of the first hierarchical time t1 = x. It is known that poles xi and matrix residues at the poles ρiαβ=aiαbiβ of such solutions as functions of the time t2 move as particles of spin generalization of the elliptic Calogero–Moser model (elliptic Gibbons–Hermsen model). In this paper, we establish the correspondence with the spin elliptic Calogero–Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the kth hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian Hk obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero–Moser system with coordinates xi and spin degrees of freedom aiα,biβ.
Tamar Friedmann, Quincy Webb
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0030945

Abstract:
We generalize the derivation of the Wallis formula for π from a variational computation of the spectrum of the hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as 2 as well as some nested radicals. We also derive Euler’s reflection formula for reciprocals of positive even integers. We show that Bohr’s correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.
W. Schlag
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0042767

Abstract:
This paper introduces some of the basic mechanisms relating the behavior of the spectral measure of Schrödinger operators near zero energy to the long-term decay and dispersion of the associated Schrödinger and wave evolutions. These principles are illustrated by means of the author’s work on decay of Schrödinger and wave equations under various types of perturbations, including those of the underlying metric. In particular, we consider local decay of solutions to the linear Schrödinger and wave equations on curved backgrounds that exhibit trapping. A particular application is waves on a Schwarzschild black hole spacetime. We elaborate on Price’s law of local decay that accelerates with the angular momentum, which has recently been settled by Hintz, also in the much more difficult Kerr black hole setting. While the author’s work on the same topic was conducted ten years ago, the global semiclassical representation techniques developed there have recently been applied by Krieger, Miao, and the author [“A stability theory beyond the co-rotational setting for critical wave maps blow up,” arXiv:2009.08843 (2020)] to the nonlinear problem of stability of blowup solutions to critical wave maps under non-equivariant perturbations.
Lingxi Liu, Xin Zhong
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0055689

Abstract:
We study an initial boundary value problem of two-dimensional nonhomogeneous micropolar fluid equations with density-dependent viscosity and non-negative density. Applying the Desjardins interpolation inequality and delicate energy estimates, we show the global existence of a unique strong solution under the condition that ‖∇μ(ρ0)‖Lq is suitably small. Moreover, we prove that the velocity and the micro-rotational velocity converge exponentially to zero in H2 as time goes to infinity.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0045188

Abstract:
In this paper, we study the space–time decay rate of solutions to three-dimensional incompressible MHD equations with Hall and ion-slip effects in the whole space R3. Based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates, we obtain the higher order mixed spatial and time derivative estimates for such a system.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0041967

Abstract:
We consider a problem of separation of variables for the Lax-integrable Hamiltonian systems governed by gl(n) ⊗ gl(n)-valued classical r-matrices r(u, v). We report on a class of classical non-skew-symmetric non-dynamical gl(n) ⊗ gl(n)-valued r-matrices rJ(u, v) labeled by arbitrary anisoropy matrix J ∈ gl(n) for which the “magic recipe” of Sklyanin [Prog. Theor. Phys., 118, 35 (1995)] in the theory of variable separation is applicable. An example of n = 3 corresponding to gl(3) ⊗ gl(3)-valued r-matrices is elaborated in detail. For the case of the r-matrices rJ(u, v) and n = 3, the coordinates of separation, the reconstruction formulas, and the Abel-type equations are explicitly written for the different types of matrices J.
J. Káninský
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0038814

Abstract:
This work builds on an existing model of discrete canonical evolution and applies it to the case of a linear dynamical system, i.e., a finite-dimensional system with vector configuration space and linear equations of motion. The system is assumed to evolve in discrete time steps. The most distinctive feature of the model is that the equations of motion can be irregular. After an analysis of the arising constraints and the symplectic form, we introduce adjusted coordinates on the phase space, which uncover its internal structure and result in a trivial form of the Hamiltonian evolution map. For illustration, the formalism is applied to the example of a massless scalar field on a two-dimensional spacetime lattice.
, , Holger Boche
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0038083

Abstract:
Communication over a quantum broadcast channel with cooperation between the receivers is considered. The first form of cooperation addressed is classical conferencing, where receiver 1 can send classical messages to receiver 2. Another cooperation setting involves quantum conferencing, where receiver 1 can teleport a quantum state to receiver 2. When receiver 1 is not required to recover information and its sole purpose is to help the transmission to receiver 2, the model reduces to the quantum primitive relay channel. The quantum conferencing setting is intimately related to quantum repeaters as the sender, receiver 1, and receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively. We develop lower and upper bounds on the capacity region in each setting. In particular, the cutset upper bound and the decode-forward lower bound are derived for the primitive relay channel. Furthermore, we present an entanglement-formation lower bound, where a virtual channel is simulated through the conference link. At last, we show that as opposed to the multiple access channel with entangled encoders, entanglement between decoders does not increase the classical communication rates for the broadcast dual.
, Xiaoning Wu, Naqing Xie
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0050978

Abstract:
Penrose et al. investigated the physical incoherence of the spacetime with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified spacetimes regarding timelike connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity.
, M. de la Rosa,
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0041384

Abstract:
In this paper, we analyze trajectories of spacelike curves that are critical points of a Lagrangian depending on its total torsion. We focus on two important families of spacetimes, generalized Robertson–Walker and standard static spacetimes. For the former, we show that such trajectories are those with a constant curvature. For the latter, we also obtain a characterization in terms of the curvature of the trajectory but in this case measured with an appropriate conformal metric.
T. Tlas
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0050011

Abstract:
A simple construction of a class of Euclidean invariant, reflection positive measures on a compactification of the space of distributions is given. An unusual feature is that the regularizations used are not reflection positive.
Journal of Mathematical Physics, Volume 62; doi:10.1063/5.0044343

Abstract:
The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.
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