Volume 57, Issue 10, October 2016

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the JacobiMaupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar threebody problem with both Newtonian and attractive inversesquare potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inversesquare potential with zero energy E. The geodesic flow on the full configuration space ℂ^{3} (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ^{2} and shape space ℝ^{3} (as well as 𝕊^{3} and the shape sphere 𝕊^{2} for the inversesquare potential when E = 0). The corresponding Riemannian submersions are described explicitly in “Hopf” coordinates which are particularly adapted to the isometries. For equal masses subject to inversesquare potentials, Montgomery shows that the zeroenergy “pair of pants” JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on ℂ^{2}, ℝ^{3}, and 𝕊^{3} with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits, and observe that the geodesic reformulation “regularizes” pairwise and triple collisions on ℂ^{2} and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on ℂ^{2} is strictly negative though it could have either sign on ℝ^{3}. However, unlike for the inversesquare potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.
 ARTICLES

 Partial Differential Equations

Orbital stability of spatially synchronized solitary waves of an mcoupled nonlinear Schrödinger system
View Description Hide DescriptionIn this paper, we investigate the orbital stability of solitarywave solutions for an mcoupled nonlinear Schrödinger system where m ≥ 2, uj are complexvalued functions of (x, t) ∈ ℝ^{2}, bjj ∈ ℝ, j = 1, 2, …, m, and bij, i ≠ j are positive coupling constants satisfying bij = bji. It will be shown that spatially synchronized solitarywave solutions of the mcoupled nonlinear Schrödinger system exist and are orbitally stable. Here, by synchronized solutions we mean solutions in which the components are proportional to one another. Our results completely settle the question on the existence and stability of synchronized solitary waves for the mcoupled system while only partial results were known in the literature for the cases of m ≥ 3 heretofore. Furthermore, the conditions imposed on the symmetric matrix B = (bij) satisfied here are both sufficient and necessary for the mcoupled nonlinear Schrödinger system to admit synchronized groundstate solutions.

Some remarks on the nonlinear Schrödinger equation with fractional dissipation
View Description Hide DescriptionWe consider the Cauchy problem for the L ^{2}critical focussing nonlinear Schrödinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup dynamics with the loglog blowup speed for .

Singular reduction modules of differential equations
View Description Hide DescriptionThe notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an indepth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and secondorder quasilinear equations are studied. Reductions of differential equations to algebraic equations and to firstorder ordinary differential equations are considered in detail within the framework proposed and are related to previous nogo results on nonclassical symmetries.
 Representation Theory and Algebraic Methods

Hopf algebras of rooted forests, cocyles, and free RotaBaxter algebras
View Description Hide DescriptionThe Hopf algebra and the RotaBaxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the “baby model” of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free RotaBaxter algebra with the structure of a cocycle Hopf algebra.

Partial classification of modules for the algebra of skewderivations over the ddimensional torus
View Description Hide DescriptionFor the algebra of skewderivations over a torus, we classify the irreducible weight modules with some restrictions.

On the Gaudin model associated to Lie algebras of classical types
View Description Hide DescriptionWe derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finitedimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.

Gradings on the real form 𝔢6,−26
View Description Hide DescriptionWe describe four fine gradings on the real form 𝔢6,−26 of the complex Lie algebra 𝔢6. They are precisely the gradings whose complexifications are fine gradings on the complex algebra. The universal grading groups are , , , and .
 Quantum Mechanics

Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions
View Description Hide DescriptionThe quantum oscillator and KeplerCoulomb problems in ddimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shapeinvariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and KeplerCoulomb potentials in curved space. The oscillator on the sphere and the KeplerCoulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the KeplerCoulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

Exact theory and numeric results for short pulse ionization of simple model atom in one dimension
View Description Hide DescriptionOur exact theory for continuous harmonic perturbation of a one dimensional model atom by parametric variations of its potential is generalized for the cases when (a) the atom is exposed to short pulses of an external harmonic electric field and (b) the forcing is represented by short bursts of different shape changing the strength of the binding potential. This work is motivated not only by the wide use of laser pulses for atomic ionization, but also by our earlier study of the same model which successfully described the ionization dynamics in all orders, i.e., the multiphoton processes, though being treated by the nonrelativistic Schrödinger equation. In particular, it was shown that the bound atom cannot survive the excitation of its potential caused by any nonzero frequency and amplitude of the continuous harmonic forcing. Our present analysis found important laws of the atomic ionization by short pulses, in particular the efficiency of ionizing this model system and presumably real ones as well.

Generalized quantum nonlinear oscillators: Exact solutions and rational extensions
View Description Hide DescriptionWe construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators. Our method is based on a connection between nonlinear oscillator systems and Schrödinger models for certain hyperbolic potentials. The rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through Jacobi type Xm exceptional orthogonal polynomials.
 Quantum Information and Computation

Correlation detection and an operational interpretation of the Rényi mutual information
View Description Hide DescriptionA variety of new measures of quantum Rényi mutual information and quantum Rényi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the Rényi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternative hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a “useless” channel whose output is decoupled from the environment. Similarly, we establish an operational interpretation of Rényi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of Rényi mutual information and Rényi conditional entropy.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Zeta functions of the Dirac operator on quantum graphs
View Description Hide DescriptionWe construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrödinger operators on graphs.
 General Relativity and Gravitation

On pseudohyperkähler prepotentials
View Description Hide DescriptionAn explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of Sp1(ℂ) × ℂ^{4n} onto the set HKp,q of local isometry classes of real analytic pseudohyperkähler metrics of signature (4p, 4q) in dimension 4n is constructed. The holomorphic functions, called prepotentials, are analogues of Kähler potentials for Kähler metrics and provide a complete parameterisation of HKp,q. In particular, there exists a bijection between HKp,q and the set of equivalence classes of prepotentials. This affords the explicit construction of pseudohyperkähler metrics from specified prepotentials. The construction generalises one due to Galperin, Ivanov, Ogievetsky, and Sokatchev. Their work is given a coordinatefree formulation and complete, selfcontained proofs are provided. The Appendix provides a vital tool for this construction: a reformulation of real analytic Gstructures in terms of holomorphic frame fields on complex manifolds.

On the Weyl and Ricci tensors of Generalized RobertsonWalker spacetimes
View Description Hide DescriptionWe prove theorems about the Ricci and the Weyl tensors on Generalized RobertsonWalker spacetimes of dimension n ≥ 3. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen’s vector, and any of the two conditions is necessary and sufficient for the Generalized RobertsonWalker (GRW) spacetime to be a quasiEinstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for RobertsonWalker spacetimes is given, in terms of Chen’s vector. In n = 4, a GRW spacetime with harmonic Weyl tensor is a RobertsonWalker spacetime.
 Dynamical Systems

On the existence of Sobolev quasiperiodic solutions of multidimensional nonlinear beam equation
View Description Hide DescriptionIn this paper, we prove the existence of quasiperiodic solutions of the multidimensional nonlinear beam equation with finitely differentiable nonlinearities and quasiperiodic forcing in time.

Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
View Description Hide DescriptionContact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper, contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.
 Classical Mechanics and Classical Fields

Curvature and geodesic instabilities in a geometrical approach to the planar threebody problem
View Description Hide DescriptionThe Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the JacobiMaupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar threebody problem with both Newtonian and attractive inversesquare potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inversesquare potential with zero energy E. The geodesic flow on the full configuration space ℂ^{3} (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ^{2} and shape space ℝ^{3} (as well as 𝕊^{3} and the shape sphere 𝕊^{2} for the inversesquare potential when E = 0). The corresponding Riemannian submersions are described explicitly in “Hopf” coordinates which are particularly adapted to the isometries. For equal masses subject to inversesquare potentials, Montgomery shows that the zeroenergy “pair of pants” JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on ℂ^{2}, ℝ^{3}, and 𝕊^{3} with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits, and observe that the geodesic reformulation “regularizes” pairwise and triple collisions on ℂ^{2} and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on ℂ^{2} is strictly negative though it could have either sign on ℝ^{3}. However, unlike for the inversesquare potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.
 Statistical Physics

On the ground state energy of the deltafunction Fermi gas
View Description Hide DescriptionThe weak coupling asymptotics to order γ of the ground state energy of the deltafunction Fermi gas, derived heuristically in the literature, is here made rigorous. Further asymptotics are in principle computable. The analysis applies to the Gaudin integral equation, a method previously used by one of the authors for the asymptotics of large Toeplitz matrices.

Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes
View Description Hide DescriptionThe Bessel process with parameter D > 1 and the Dyson model of interacting Brownian motions with coupling constant β > 0 are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi’s theta functions. They are called the elliptic Bessel process, eBES^{(D)}, and the elliptic Dyson model, eDYS^{(β)}, respectively. Both are realized on the circumference of a circle [0, 2πr) with radius r > 0 as temporally inhomogeneous processes defined in a finite time interval [0, t ∗), t ∗ < ∞. Transformations of them to Schrödingertype equations with timedependent potentials lead us to proving that eBES^{(D)} and eDYS^{(β)} can be constructed as the timedependent Girsanov transformations of Brownian motions. In the special cases where D = 3 and β = 2, observables of the processes are defined and the processes are represented for them using the Brownian paths winding round a circle and pinned at time t ∗. We show that eDYS^{(2)} has the determinantal martingale representation for any observable. Then it is proved that eDYS^{(2)} is determinantal for all observables for any finite initial configuration without multiple points. Determinantal processes are stochastic integrable systems in the sense that all spatiotemporal correlation functions are given by determinants controlled by a single continuous function called the spatiotemporal correlation kernel.

Novel dissipative properties of the master equation
View Description Hide DescriptionRecent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and nonadiabatic parts, where the nonadiabatic part is also known as the free energy dissipation rate. In this paper, we present some nonzero lower bounds for the free energy, the entropy production rate, and its adiabatic and nonadiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for nonequilibrium dynamics which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also give a mathematical application of the nonzero lower bounds by studying the longtime behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallistype free energy and its dissipation rate.