ALBERT

Description: The Wegner estimate for the Hamiltonian of the Anderson model for the special Gaussian random magnetic field is extended to more general magnetic fields. The Lifshitz tail upper bounds of the integrated density of states as analyzed by Nakamura are reviewed and extended so that Gaussian random magnetic fields can be treated. By these and multiscale analysis, the Anderson localization at low energies is proven.

Description: Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie enveloping algebra and support representations of semigroups. We explicitly construct here basis dependent RHS of the line and half-line and relate them to the universal enveloping algebras of the Weyl-Heisenberg algebra and su (1, 1), respectively. The complete sub-structure of both RHS and of the operators acting on them is obtained from their algebraic structures or from the related fractional Fourier transforms. This allows us to describe both quantum and signal processing states and their dynamics. Two relevant improvements are introduced: (i) new kinds of filters related to restrictions to subspaces and/or the elimination of high frequency fluctuations and (ii) an operatorial structure that, starting from fix objects, describes their time evolution.

Description: We consider the p -Laplacian problem − ε p Δ p u + V ( x ) u p − 2 u = f ( u ) , u ∈ W 1 , p ( R N ) , where p ∈ (1, N ) and f ( s ) is of critical growth. In this paper, we construct a single peak solution around an isolated component of the positive local minimum points of V as ε → 0 with a general nonlinearity f . In particular, the monotonicity of f ( s )/ s p −1 and the so-called Ambrosetti-Rabinowitz condition are not required.

Description: It is shown that the initial value problem for an integrable Novikov system is well-posed in Sobolev spaces H s , s 〉 3/2, in the sense of Hadamard. Furthermore, it is proved that the dependence on initial data is sharp, i.e., the data-to-solution map is continuous but not uniformly continuous. Also, peakon traveling wave solutions are used to prove that the solution map is not uniformly continuous in H s for s 〈 3/2.

Description: Our discussion here mainly focuses on the formation of singularities for solutions to the N -dimensional Euler-Poisson equations with attractive forces, in radial symmetry. Motivated by the integration method of Yuen, we prove two blow-up results under the conditions that the solutions have compact radius R ( t ) or have no compact support restriction, which generalize the ones Yuen obtained in 2011 [M. W. Yuen, “Blowup for the C 1 solution of the Euler-Poisson equations of gaseous stars in R N ,” J. Math. Anal. Appl. 383 , 627-633 (2011)].

Description: In this paper, we provide a new method for establishing the blowup of C 2 solutions for the pressureless Euler-Poisson system with attractive forces for R N ( N ≥ 2) with ρ (0, x 0 ) 〉 0 and Ω 0 i j ( x 0 ) = 1 2 ∂ i u j ( 0 , x 0 ) − ∂ j u i ( 0 , x 0 ) = 0 at some point x 0 ∈ R N . By applying the generalized Hubble transformation   div   u ( t , x 0 ( t ) ) = N a ̇ ( t ) a ( t ) to a reduced Riccati differential inequality derived from the system, we simplify the inequality into the Emden equation a ̈ ( t ) = − λ a ( t ) N − 1 , a ( 0 ) = 1 , a ̇ ( 0 ) =   div   u ( 0 , x 0 ) N . Known results on its blowup set allow us to easily obtain the blowup conditions of the Euler-Poisson system.

Description: Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated.

Description: The paper focuses on studying the Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. First, the Hamilton equations of nonholonomic nonconservative systems on time scales are established, which is based on the Lagrange equations for nonholonomic systems on time scales. Then, based upon the quasi-invariance of Hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale, the Noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained. Finally, an example is presented to illustrate the application of the results.

Description: General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article, I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by the demanding invariance of the laws of physics to the change of coefficient structures in cohomology.

Description: The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.

Description: The BPS Skyrme model is a model containing an SU (2)-valued scalar field, in which a Bogomol’nyi-type inequality can be satisfied by soliton solutions (skyrmions). In this model, the energy density of static configurations is the sum of the square of the topological charge density plus a potential. The topological charge density is nothing else but the pull-back of the Haar measure of the group SU (2) on the physical space by the field configuration. As a consequence, this energy expression has a high degree of symmetry: it is invariant to volume preserving diffeomorphisms both on physical space and on the target space. We demonstrate here that in the BPS Skyrme model such solutions exist that a fraction of its charge and energy densities is localised, and the remaining part can be far away, not interacting with the localised part.

Description: Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. There exist very few results for steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes due to the technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loève expansion of complex-valued Ornstein-Uhlenbeck process.

Description: Using an appropriate norm on the space of entire functions, we extend to the complex plane the renormalization group method as developed by Bricmont et al. The method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems. The time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical N -vector model. We show that, for initial conditions belonging to a certain Fréchet space of entire functions of exponential type, the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions.

Description: In this paper, we study the existence of positive solution for the following class of fractional elliptic equation ϵ 2 s ( − Δ ) s u + V ( z ) u = λ u q − 2 u + u 2 s ∗ − 2 u in R N , where ϵ , λ 〉 0 are positive parameters, q ∈ ( 2 , 2 s ∗ ) , 2 s ∗ = 2 N N − 2 s , N 〉 2 s , s ∈ ( 0 , 1 ) , ( − Δ ) s u is the fractional Laplacian, and V is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.

Description: We consider a system of equations governing the motion of a viscous, compressible, and heat conducting liquid-like fluid, with a general equation of state (EOS) of Mie-Grüneisen type. In addition, we suppose that the viscosity coefficients may decay to zero for large values of the temperature. We show the existence of global-in-time weak solution, derive a relative energy inequality, and compare the weak solutions with strong one emanating from the same initial data—the weak strong uniqueness property.

Description: In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α -models of turbulence and therefore are called the α -regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α -regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al. , Nonlinearity 21 , 879–898 (2008); Cao et al. , Numer. Funct. Anal. Optim. 30 , 46–69 (2009)] on the non-viscous α -regularization of the nonlinear Schrödinger equation.

Description: We study an important system of Schrödinger equations with linear and nonlinear couplings arising from Bose-Einstein condensates. We use the Nehari manifold to prove the existence of a ground state solution; moreover, we give the sign of the solutions depending on linear coupling; by using index theory and Nehari manifold, we prove that there exist infinitely many positive bound state solutions.

Description: We consider Landau Hamiltonians with a weak coupling random electric potential of breather type. Under appropriate assumptions we prove a Wegner estimate. It implies the Hölder continuity of the integrated density of states. The main challenge is the problem how to deal with non-linear dependence on the random parameters.

Description: We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ) 1/2 , whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into non-overlapping, orbitally labelled E ( k , l ) series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive l th series eigenvalues. Each of them is 2 l + 1-degenerate. The l = 0 eigenvalues series E ( k ,0) are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E ( k ,0) ( d = 3) = E 2 k ( d = 1). Likewise, the eigenfunctions ψ ( k ,0) ( d = 3) and ψ 2 k ( d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).

Description: The aim of this paper is to prove the existence of a new symmetric family of periodic solutions of the generalized van der Waals Hamiltonian. In fact, we prove the existence of several families of first kind symmetric periodic solutions as continuation of circular orbits of the Kepler problem in the spatial case.

Description: The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ d 1 ⊗ ℂ d 2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.

Description: The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network and, more specifically, as a linear map from the input space to the output space. The quantum max-flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min-cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.

Description: In this paper, we establish the eigenvalue asymptotics for non-self-adjoint Dirac–Bessel operators on (0, 1) with arbitrary real angular momenta and square integrable potentials, which gives the first step for solution of the related inverse problem. The approach is based on a careful examination of the corresponding characteristic functions and their zero distribution.

Description: An adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase σ ( r → ) of a wave function exp i ħ σ ( r → ) and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.

Description: Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states ( ρ or σ ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n ) identical copies of the state ( ρ ⊗ n or σ ⊗ n ) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability α n of erroneously inferring the state to be σ , the probability β n of erroneously inferring the state to be ρ decays exponentially in n , with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.

Description: We consider two non-interacting infinite quantum spin chains immersed in a common thermal environment and undergoing a local dissipative dynamics of Lindblad type. We study the time evolution of collective mesoscopic quantum spin fluctuations that, unlike macroscopic mean-field observables, retain a quantum character in the thermodynamical limit. We show that the microscopic dissipative dynamics is able to entangle these mesoscopic degrees of freedom, through a purely mixing mechanism. Further, the behaviour of the dissipatively generated quantum correlations between the two chains is studied as a function of temperature and dissipation strength.

Description: In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

Description: A functional realization of the Lie algebra s l 3 , R as a Vessiot–Guldberg–Lie algebra of second order differential equation (SODE) Lie systems is proposed. It is shown that a minimal Vessiot–Guldberg–Lie algebra L V G is obtained from proper subalgebras of s l 3 , R for each of the SODE Lie systems of this type by particularization of one functional and two scalar parameters of the s l 3 , R -realization. The relation between the various Vessiot–Guldberg–Lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of SODE Lie systems.

Description: We introduce a generalization of the Jaynes-Cummings model and study some of its properties. We obtain the energy spectrum and eigenfunctions of this model by using the tilting transformation and the squeezed number states of the one-dimensional harmonic oscillator. As physical applications, we connect this new model to two important and novelty problems: the relativistic parametric amplifier and the quantum simulation of a single trapped ion.

Description: We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in Euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.

Description: We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.

Description: We study the constrained Ostrogradski-Hamilton framework for the equations of motion provided by mechanical systems described by second-order derivative actions with a linear dependence in the accelerations. We stress out the peculiar features provided by the surface terms arising for this type of theories and we discuss some important properties for this kind of actions in order to pave the way for the construction of a well defined quantum counterpart by means of canonical methods. In particular, we analyse in detail the constraint structure for these theories and its relation to the inherent conserved quantities where the associated energies together with a Noether charge may be identified. The constraint structure is fully analyzed without the introduction of auxiliary variables, as proposed in recent works involving higher order Lagrangians. Finally, we also provide some examples where our approach is explicitly applied and emphasize the way in which our original arrangement results in propitious for the Hamiltonian formulation of covariant field theories.

Description: Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincaré duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree k of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincaré duality for the new cohomology groups.

Description: We introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A , by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8 , 1–145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) n -Nambu-Poisson algebras over an algebraically closed field of characteristic zero.

Description: The higher-order superintegrability of the two-dimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational ratio of frequencies is directly related with the existence of some complex functions with interesting Poisson bracket properties. First the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi-Hamiltonian complex structure. In the second part several real symplectic structures are obtained and the properties of the recursion operators are studied.

Description: We study finite W -algebras associated to even regular (principal) nilpotent elements for the family of simple exceptional Lie superalgebras D (2, 1; α ) and for the universal central extension of

Description: We investigate the spectrum of the self-similar Laplacian, which generates the so-called “ pq random walk” on the integer half-line ℤ + . Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever p ≠ 1 2 . This serves as a toy model for generating singularly continuous spectrum, which can be generalized to more complicated settings. We hope it will provide more insight into Fibonacci-type and other weakly self-similar models.

Description: In this paper, we study Lie super-bialgebra and quantization of the super Virasoro algebra, whose even part is the centerless twisted Heisenberg-Virasoro algebra. By using some Liu-Pei-Zhu’s results, we prove that all Lie super-bialgebra structures on the super Virasoro algebra are triangular coboundary. Furthermore, we quantize the super Virasoro algebra by the Drinfel’d twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.

Description: The paper is motivated by the importance of the rephasing invariance of the CKM (Cabibbo-Kobayashi-Maskawa) matrix observables. These observables appear in the discussion of the CP violation in the standard model (Jarlskog invariant) and also in the renormalization group equations for the quark Yukawa couplings. Our discussion is based on the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the main theorem we show that a general rephasing invariant monomial can be expressed as a product of at most five factors: four of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements. We also show that the imaginary part of any rephasing invariant monomial is proportional to the Jarlskog’s invariant J or is 0.

Description: In this work we study the dimensional reduction of smooth circle invariant Yang-Mills instantons defined on 4-manifolds which asymptotically become circle fibrations over hyperbolic 3-space. A suitable choice of the 4-manifold metric within a specific conformal class gives rise to singular and smooth hyperbolic monopoles. A large class of monopoles is obtained if the conformal factor satisfies the Helmholtz equation on hyperbolic 3-space. We describe simple configurations and relate our results to the Jackiw-Nohl-Rebbi construction, for which we provide a geometric interpretation.

Description: We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53 , 095203 (2012)] that such a transition occurs when the angular velocity is of order ε −4 , with ε −2 denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ω c such that if Ω = Ω 0 /ε 4 with Ω 0 〉 Ω c , the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.

Description: Highest weight modules for U q ( g l 2 ̂ ) are endowed with a structure of modules for the quantum toriodal algebra U κ of type sl 2 . Using this, we define U κ actions on the space of vertex operators for irreducible highest weight U q ( g l 2 ̂ ) modules. Highest or lowest weight vectors of the thus obtained U κ modules are expressed in terms of an intertwiner for U q ( s l 2 ̂ ) modules and an extra boson. The submodules generated by these vectors are investigated.

Description: We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

Description: An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, C ̆ encov, and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter h . Due to the sum rules for the frequency moments of the dynamic structure factor, new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator S are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.

Description: In physics and in mathematics Z 2 n -gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the “scalar product” of the involved Z 2 n -degrees. The Z 2 n -supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define Z 2 n -supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z 2 • -supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n -fold vector bundle has a canonical “superization” to a Z 2 n -supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z 2 n -context.

Description: To each hyperbolic Landau level of the Poincaré disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its atomic decomposition as a perturbation of the negative binomial distribution by a finitely supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we derive a Lévy-Khintchine-type representation of its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-Lévy measure, we introduce a new infinitely divisible distribution for which we derive the characteristic function.

Description: In this work, we analytically study the Schrödinger equation for the (non-pure) dipolar ion potential V ( r ) = q / r + D cos θ / r 2 , in the case of 2D systems (systems in two-dimensional Euclidean plane) using the separation of variables and the Mathieu equations for the angular part. We give the expressions of eigenenergies and eigenfunctions and study their dependence on the dipole moment D . Imposing the condition of reality on the energies E n , m implies that the dipole moment must not exceed a maximum value, otherwise the corresponding bound state disappears. We also find that the s states ( m = 0) can no longer exist in the system as soon as the dipole term is present.

Description: We begin showing that for even dimensional vector spaces V all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of V are restrictions to V of inner automorphisms of the algebra. Thus under orthogonal transformations P and T —space and time reversal—all algebra elements, including vectors v and spinors φ , transform as v → xvx −1 and φ → xφx −1 for some algebra element x . We show that while under combined PT spinor φ → xφx −1 remains in its spinor space, under P or T separately φ goes to a different spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.

Description: We explore some of the cosmological implications of the recent classical nonlocal generalization of Einstein’s theory of gravitation in which nonlocality is due to the gravitational memory of past events. In the Newtonian regime of this theory, the nonlocal character of gravity simulates dark matter in spiral galaxies and clusters of galaxies. However, dark matter is considered indispensable as well for structure formation in standard models of cosmology. Can nonlocal gravity solve the problem of structure formation without recourse to dark matter? Here we make a beginning in this direction by extending nonlocal gravity in the Newtonian regime to the cosmological domain. The nonlocal analog of the Zel’dovich solution is formulated and the consequences of the resulting nonlocal Zel’dovich model are investigated in detail.

Description: We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S = 1 2 , the model is the Heisenberg ferromagnet, and for general spin S ∈ 1 2 N , it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S = 1 2 ). The critical temperature is shown to coincide (as a function of S ) with that of the q = 2 S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.

Description: In this paper, the inverse scattering transform associated with a Riemann-Hilbert problem is formulated for the FQXL model: a generalized Camassa-Holm equation m t = 1 2 k 1 [ m ( u 2 − u x 2 ) ] x + 1 2 k 2 ( 2 m u x + m x u ) , m = u − u x x , which was originally included in the work of Fokas [Physica D 87 , 145 (1995)] and was recently shown to be integrable in the sense of Lax pair, bi-Hamilton structure, and conservation laws by Qiao, Xia, and Li [e-print arXiv:1205.2028v2 (2012)]. We have discussed the following properties: direct scattering problems and Jost solutions, asymptotical and analytical behavior of Jost solutions, the scattering equations in a Riemann-Hilbert problem, and the multi-soliton solutions of the FQXL model. Then, one-soliton and two-soliton solutions are presented in a parametric form as a special case of multi-soliton solutions.

Description: We compare the known in literature, explicitly covariant 4-dimensional formula for the symmetric energy-momentum tensor of electromagnetic field in a medium and the energy-momentum tensor derived by Abraham in the 3-dimensional vector form. It is shown that these two objects coincide only on the physical configuration space Γ ¯ , formed by the field vectors and the velocity of the medium, which satisfy the Minkowski constitutive relations. It should be emphasized that the 3-dimensional vector formulae for the components of the energy-momentum tensor were obtained by Abraham only on Γ ¯ , and the task of their extension to the whole unconditional configuration space Γ was not posed. In order to accomplish the comparison noted above, we derive the covariant formula a new by another method, namely, by generalizing the Abraham reasoning. The comparison conducted enables one to treat the explicitly covariant formula as a unique consistent extension of the Abraham formulae to the whole configuration space Γ. Thus the question concerning the relativistic covariance of the original 3-dimensional Abraham formulae defined on Γ ¯ is solved positively. We discuss in detail the relativistic covariance of the 3-dimensional vector formulae for individual components of the 4-dimensional tensors in electrodynamics which is manifested in the form-invariance of these formulae under Lorentz transformations.

Description: In this paper, we consider the following quasilinear elliptic equation with Hardy potential and Dirichlet boundary condition: − ∑ i , j = 1 N D j ( a i j ( x , u ) D i u ) + 1 2 ∑ i , j = 1 N D s a i , j ( x , u ) D i u D j u − λ | x | − 2 u = f ( x , u ) i n Ω , where Ω ⊂ ℝ N ( N ≥ 3) is a smooth bounded domain, D i = ∂ ∂ x i , D s a i j ( x , s ) = ∂ ∂ s a i j ( x , s ) , and 0 ≤ λ 〈 λ ∗ : = ( N − 2 2 ) 2 , and λ | x | −2 is called the Hardy potential. By using the perturbation method, we prove the existence of infinitely many solutions for the above problem.

Description: We present a method to compute the Laurent expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation ε. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.

Description: Considering a 3 + 1 dimensional lattice quantum chromodynamics (QCD) model defined with the improved Wilson action, three flavors, and 4 × 4 Dirac spin matrices, in the strong coupling regime, we reanalyze the question of the existence of the eightfold way baryons and complete our previous work where the existence of isospin octet baryons was rigorously solved. Here, we show the existence of isospin decuplet baryons which are associated with isolated dispersion curves in the subspace of the underlying quantum mechanical Hilbert space with vectors constructed with an odd number of fermion and antifermion basic quark and antiquark fields. Moreover, smoothness properties for these curves are obtained. The present work deals with a case for which the traditional method to solve the implicit equation for the dispersion curves, based on the use of the analytic implicit function theorem, cannot be applied. We do not have only one but two solutions for each one-baryon decuplet sector with fixed spin third component. Instead, we apply the Weierstrass preparation theorem, which also provides a general method for the general degenerate case. This work is completed by analyzing a spectral representation for the two-baryon correlations and providing the leading behaviors of the field strength normalization and the mass of the spectral contributions with more than one-particle. These are needed results for a rigorous analysis of the two-baryon and meson-baryon particle spectra.

Description: We define the 3-point Virasoro algebra and construct a representation of it on a previously defined Fock space for the 3-point affine algebra.

Description: We analyze entropic uncertainty relations for two orthogonal measurements on a N -dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.

Description: This paper constructs a nonlinear filtering framework that admits appearances of new information processes at random times by introducing piecewise enlargements of filtrations and proposes a new energy-based Schrodinger evolution expressed as a stochastic differential equation on a complex Hilbert space. Each information process is modeled as the sum of a random variable taking the eigenvalues of a Hamiltonian and an independent Brownian bridge noise. It is shown that under a piecewise enlarged filtration, the wave function is a jump-diffusion process until it collapses at some terminal time. In between discontinuities, the dynamics of the state vector are governed by different Wiener processes and diffusion coefficients. This motivates the introduction of an inclusive chain of Kolmogorov probability spaces or a *-isomorphic chain of commutative von Neumann probability spaces, on which the quantum system evolves differently based on the number of active information processes. The expectation of the Hamiltonian at a given state is the solution of a second-order nonlinear differential equation determined by one of the possible regimes that the quantum system belongs to. It is shown that the collapse rate is a submartingale with positive jumps and the Shannon entropy process is a supermartingale with expected negative jumps when passing to higher-order probability spaces. The framework is extended to the case when the Hamiltonian is modeled as a function of a set of commutative operators, where each operator is associated with a different piecewise enlarged filtration.

Description: The important problem of how to prepare a quantum mechanical system, S , in a specific initial state of interest—e.g., for the purposes of some experiment—is addressed. Three distinct methods of state preparation are described. One of these methods has the attractive feature that it enables one to prepare S in a preassigned initial state with certainty, i.e., the probability of success in preparing S in a given state is unity. This method relies on coupling S to an open quantum-mechanical environment, E , in such a way that the dynamics of S ∨ E pulls the state of S towards an “attractor,” which is the desired initial state of S . This method is analyzed in detail.

Description: Conformal Galilei algebras (CGAs) labeled by d , ℓ (where d is the number of space dimensions and ℓ denotes a spin-ℓ representation w.r.t. the

Description: The van der Pol-Mathieu-Duffing equation x ̈ + ( Ω 0 2 + h 1 cos Ω 1 t + h 2 cos Ω 2 t ) x − ( α − β x 2 ) x ̇ − h 3 x 3 = h 4 Ω 3 2 cos x cos Ω 3 t is considered in this paper, where α , β , h 1 , h 2 , h 3 , h 4 , Ω 1 , Ω 2 are small parameters, α , β 〉 0, the frequency Ω 3 is large compared to Ω 1 and Ω 2 , the above parameters are real. For ∀ α , β 〉 0, we use KAM (Kolmogorov-Arnold-Moser) theory to prove that the van der Pol-Mathieu-Duffing equation possesses quasi-periodic solutions for most of the parameters Ω 0 , Ω 1 , Ω 2 , Ω 3 , it verifies some phenomenon of Fahsi and Belhaq [Commun. Nonlinear Sci. 14 , 244-253 (2009)] and can be regarded as a extension of Abouhazim et al. [Nonlinear Dyn. 39 , 395-409 (2005)].

Description: We carry out a systematic qualitative analysis of the two quadratic schemes of generalized oscillators recently proposed by Quesne [J. Math. Phys. 56 , 012903 (2015)]. By performing a local analysis of the governing potentials, we demonstrate that while the first potential admits a pair of equilibrium points one of which is typically a center for both signs of the coupling strength λ , the other points to a centre for λ 〈 0 but a saddle λ 〉 0. On the other hand, the second potential reveals only a center for both the signs of λ from a linear stability analysis. We carry out our study by extending Quesne’s scheme to include the effects of a linear dissipative term. An important outcome is that we run into a remarkable transition to chaos in the presence of a periodic force term f cos ωt .

Description: Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main theme in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallelly transported along geodesics. It is also shown that the number of linear independent rank- p antisymmetric affine tensor fields in n -dimensions is bounded by ( n + 1)!/ p !( n − p )!. We also derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, we discuss the existence of antisymmetric affine tensor fields on various spacetimes.

Description: In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V ( x ) = A 2 [ 1 + tanh ( x / a ) ] and a finite barrier V ( x ) = A sech 2 ( x / a ). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k -space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability T s ( k ), i.e.,  T s ( k ) displays a unit step function-like profile Θ( k − k 0 ), especially when a is large, say, a ≫ 1/ κ , 1/ k . And k 0 ≡ m A / ħ is exactly where the peak appears in phase or dwell time k -spectrum. Thus only those particles with k in a very narrow interval around k 0 are capable to dwell in the central region of the barrier sufficiently long.

Description: In this paper, we are concerned with a class of Schrödinger-Poisson systems with the asymptotically linear or asymptotically 3-linear nonlinearity. Under some suitable assumptions on V , K , a , and f , we prove the existence, nonexistence, and asymptotic behavior of solutions via variational methods. In particular, the potential V is allowed to be sign-changing for the asymptotically linear case.

Description: We address the criticism of Frewer et al. concerning the paper “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54 , 072901 (2013)]. Most importantly, we stress that we never claimed that any new statistical symmetries were found in this paper. The aim of this paper was to apply the Lie group analysis to an equation with functional derivatives and derive invariant solutions for this equation. These results still stand as they are, most important, mathematically correct. We address also other critical statements of Frewer et al. and show that there is a connection between the translational invariance of statistics and transformations of the functional Φ. To sum up, key ideas and fundamental result in the work of Wacławczyk and Oberlack are still unaffected.

Description: The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Wacławczyk and Oberlack [J. Math. Phys. 54 , 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered Burgers equation essential symmetries are not captured.

Description: We apply the formalism of quantum measurement theory to the idealized measurement of the position of a particle with an optical interferometer, finding that the backaction of counting entangled photons systematically collapses the particle’s wavefunction toward a narrow Gaussian wavepacket at the location x est determined by the measurement without appeal to environmental decoherence or other spontaneous collapse mechanism. Further, the variance in the particle’s position, as calculated from the post-measurement wavefunction, agrees precisely with shot-noise limited uncertainty of the measured x est . Both the identification of the absolute square of the particle’s initial wavefunction as the probability density for x est and the de Broglie hypothesis emerge as consequences of interpreting the intensity of the optical field as proportional to the probability of detecting a photon. Linear momentum information that is encoded in the particle’s initial wavefunction survives the measurement, and the pre-measurement expectation values are preserved in the ensemble average.

Description: We study the entropy increase of quantum systems evolving under primitive, doubly stochastic Markovian noise and thus converging to the maximally mixed state. This entropy increase can be quantified by a logarithmic-Sobolev constant of the Liouvillian generating the noise. We prove a universal lower bound on this constant that stays invariant under taking tensor-powers. Our methods involve a new comparison method to relate logarithmic-Sobolev constants of different Liouvillians and a technique to compute logarithmic-Sobolev inequalities of Liouvillians with eigenvectors forming a projective representation of a finite abelian group. Our bounds improve upon similar results established before and as an application we prove an upper bound on continuous-time quantum capacities. In the last part of this work we study entropy production estimates of discrete-time doubly stochastic quantum channels by extending the framework of discrete-time logarithmic-Sobolev inequalities to the quantum case.

Description: We investigate the Schrödinger representations of certain infinite-dimensional Heisenberg groups, using their corresponding Wigner transforms.

Description: Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynomials.

Description: In Paper II, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which gives a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. [J. Approx. Theory 204 , 1 (2016); e-print arXiv:1506.03651 [math.CA]] to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type.

Description: The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU ( N ) can be expressed as a matrix polynomial of order N − 1 in a traceless N × N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N -vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an N − 1 -simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.

Description: In this paper, we study the large-time asymptotic behavior of contact wave for the Cauchy problem of one-dimensional compressible Navier-Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and proves that it is nonlinearly stable provided that the strength of the contact discontinuity and the perturbation of the initial data are suitably small.

Description: We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As a part of this formalism, we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e., SL (2, ℂ)) 2-spinors as well as to space (i.e., SU (2, ℂ)) 2-spinors. We compute expressions for the variations of the connection and the curvature spinors.

Description: This review article is concerned with a recently uncovered connection between operator spaces , a noncommutative extension of Banach spaces, and quantum nonlocality , a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states.

Description: We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity. We exhibit three main families of Hamiltonian models with two, three, and four degrees of freedom aiming at modeling the complexity of the bunch of particles in the Vlasov dynamics.

Description: The dynamics of a test particle (a particle with zero vorticity) advected by the velocity field of N point-vortices with vorticities Γ j ,   j = 1, … N , is considered. Making an analogy with similar studies in celestial mechanics, we call such a study a “restricted N -vortex problem” or ( N + 1)-vortex problem. In particular, we study and characterize the global planar dynamics of some restricted 3 and 4-vortex problems, as a function of the vorticities Γ j of the vortices.

Description: Using a Lagrangian framework, we study overdamping phenomena in gyroscopic systems composed of two components, one of which is highly lossy and the other is lossless. The losses are accounted by a Rayleigh dissipation function. As we have shown previously, for such a composite system, the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such system, a rather universal phenomenon of selective overdamping occurs. Namely, first of all, the high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes. Second of all, the rest of the low-loss modes remain oscillatory (i.e., the underdamped modes ), each with an extremely high quality factor (Q-factor) that actually increases as the loss of the lossy component increases. We prove that selective overdamping is a generic phenomenon in Lagrangian systems with gyroscopic forces and gives an analysis of the overdamping phenomena in such systems. Moreover, using perturbation theory, we derive explicit formulas for upper bound estimates on the amount of loss required in the lossy component of the composite system for the selective overdamping to occur in the generic case and give Q-factor estimates for the underdamped modes. Central to the analysis is the introduction of the notion of a “dual” Lagrangian system and this yields significant improvements on some results on modal dichotomy and overdamping. The effectiveness of the theory developed here is demonstrated by applying it to an electric circuit with a gyrator element and a high-loss resistor.

Description: We study the problem of coupling Einstein’s equations to a physically well-motivated relativistic modification of the Navier-Stokes equations. Under a technical condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid’s dynamic velocity has vanishing divergence, and show that the solutions enjoy the finite propagation speed property.

Description: We derive and interpret solutions of time-harmonic Maxwell’s equations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g., graphene sheet, lying between two unbounded isotropic and non-magnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmon-polariton of transverse-magnetic polarization of plane waves.

Description: Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and Lévy-Leblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.

Description: In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex PT symmetric potentials. We focus our attention on the conventional potentials such as the generalized Pöschl Teller (GPT), Scarf-I, and PT symmetric Scarf-II which are invariant under certain parametric transformations. The resulting set of potentials is shown to yield a completely different behavior of the bound state solutions. Further, the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and PT symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature. We subsequently take up a study of the newly discovered rationally extended SI potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs). We discuss the transformations of the corresponding Casimir operator employing the properties of the so (2, 1) algebra.

Description: We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of the Hadamard form, provided that the time integral of the spatial sup-norm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state.

Description: In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.

Description: In this paper, we try to understand the geometry for a nonlocal nonlinear Schrödinger equation (nonlocal NLS) and its discrete version introduced by Ablowitz and Musslimani, Phys. Rev. Lett. 110 , 064105 (2013); Phys. Rev. E 90 , 042912 (2014). We show that, under the gauge transformations, the nonlocal focusing NLS and the nonlocal defocusing NLS are, respectively, gauge equivalent to a Heisenberg-like equation and a modified Heisenberg-like equation, and their discrete versions are, respectively, gauge equivalent to a discrete Heisenberg-like equation and a discrete modified Heisenberg-like equation. Although the geometry related to the nonlocal NLS and its discrete version is not very clear, from the gauge equivalence, we can see that the properties between the nonlocal NLS and its discrete version and NLS and discrete NLS have significant difference. By constructing the Darboux transformation for discrete nonlocal NLS equations including the cases of focusing and defocusing, we derive their discrete soliton solutions, which differ from the ones obtained by using the inverse scattering transformation.

Description: In a series of papers (see Foias et al. [J. Dyn. Differ. Equations 14 (1), 1–35 (2002)] and the pertinent references therein), the 3D Navier-Stokes- α model was shown to be a useful complement to the 3D Navier-Stokes equations, and in particular, to be a good Reynolds version of the latter equations. In this work, we introduce a simple Reynolds averaging which, due to the wall roughness, transforms the Navier-Stokes equations into the Navier-Stokes- α model.

Description: Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrödinger-Virasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV ( a , b ), where a ,   b are complex numbers. The conformal derivations of CSV ( a , b ) are uniformly determined. The rank one conformal modules and ℤ-graded free intermediate series modules over CSV ( a , b ) are classified. Corresponding results of the conformal subalgebra CHV ( a , b ) of CSV ( a , b ) are also presented.

Description: In this paper, we study the long-time behavior of the solutions of non-autonomous parabolic equations with memory in cases when the nonlinear term satisfies subcritical and critical growth conditions. In order to do this, we show that the family of processes associated to original systems with heat source f ( x , t ) being translation bounded in L loc 2 ( R ; L 2 ( Ω ) ) is dissipative in higher energy space M α , 0 〈 α ≤ 1, and possesses a compact uniform attractor in M 0 .

Description: This paper considers the global regularity to the 3D incompressible MHD equations with large initial data in bounded domains. Let μ , ν , u , and b denote the viscosity coefficient, magnetic diffusivity, velocity field, and magnetic field, respectively. We construct new systems for ( u − b ) and ( u + b ) to overcome the difficulties caused by the large initial data. It is shown that ( u , b ) H 1 is globally bounded as long as ( u 0 − b 0 ) H 1 + μ − ν ( μ + ν ) − 1 or ( u 0 + b 0 ) H 1 + μ − ν ( μ + ν ) − 1 is sufficiently small, which indicates that the Navier-Stokes equations can be regularized by the magnetic field.

Description: In this paper, we study minimizers of the Hartree-type energy functional E a ( u ) ≔ ∫ R N ∇ u ( x ) 2 + V ( x ) u ( x ) 2 d x − a p ∫ R N I α ∗ u ( x ) p u ( x ) p d x , a ≥ 0 under the mass constraint ∫ R N u 2 d x = 1 , where p = N + α + 2 N with α ∈ (0, N ) for N ≥ 2 is the mass critical exponent. Here I α denotes the Riesz potential and the trapping potential 0 ≤ V ( x ) ∈ L loc ∞ ( R N ) satisfies lim x → ∞ V ( x ) = ∞ . We prove that minimizers exist if and only if a satisfies a 〈 a ∗ = Q 2 2 ( p − 1 ) , where Q is a positive radially symmetric ground state of − Δ u + u = ( I α ∗ u p ) u p − 2 u in ℝ N . The uniqueness of positive minimizers holds if a 〉 0 is small enough. The blow-up behavior of positive minimizers as a ↗ a ∗ is also derived under some general potentials. Especially, we prove that minimizers must blow up at the central point of the biggest inscribed sphere of the set Ω ≔ { x ∈ ℝ N , V ( x ) = 0} if Ω 〉 0 .

Description: We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the O ( N )-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.

Description: Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.

Description: Weyl-orbit functions have been defined for each simple Lie algebra, and permit Fourier-like analysis on the fundamental region of the corresponding affine Weyl group. They have also been discretized, using a refinement of the coweight lattice, so that digitized data on the fundamental region can be Fourier-analyzed. The discretized orbit function has arguments that are redundant if related by the affine Weyl group, while its labels, the Weyl-orbit representatives, invoke the dual affine Weyl group. Here we discretize the orbit functions in a novel way, by using the weight lattice. A cleaner theory results with symmetry between the arguments and labels of the discretized orbit functions. Orthogonality of the new discretized orbit functions is proved, and leads to the construction of unitary, symmetric matrices with Weyl-orbit-valued elements. For one type of orbit function, the matrix coincides with the Kac-Peterson modular S matrix, important for Wess-Zumino-Novikov-Witten conformal field theory.

Description: In the paper it is shown that due to separation of variables in the Laplace-Beltrami operator (Hamiltonian of a free quantum particle) in horospheric and quasi-Cartesian coordinates of three dimensional Lobachevsky space, it is possible to introduce standard (“conventional” according to Perelomov [Generalized Coherent States and Their Applications (Springer-Verlag, 1986), p. 320]) coherent states. Some problems (oscillator on horosphere, charged particle in analogy of constant uniform magnetic field) where coherent states are suitable for treating were considered.

Description: The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler’s elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.

Description: In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall “coherent state” transform for compact Lie groups G , which we prove for G = U (1) n and support by numerical evidence for G = SU (2). The reason for conjoining this conjecture with the main topic of this article originates in the observation that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups G . But, as Weyl and Berezin quantisation for ℝ 2 d are intimately related by heat kernel evolution, it is natural to ask whether a similar connection exists for compact Lie groups as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former.

Description: In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

Description: A novel class of N -body problems is identified, with N an arbitrary positive integer ( N ≥ 2). These models are characterized by Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z -plane. These highly nonlinear equations feature N arbitrary coupling constants, yet they can be solved by algebraic operations and if all the N coupling constants are real and rational the corresponding N -body problem is isochronous : its generic solutions are all completely periodic with an overall period T independent of the initial data (but many solutions feature subperiods T / p with p integer ). It is moreover shown that these models are Hamiltonian.

Description: We employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. We prove a new uniform ergodic theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness.