Volume 54, Issue 12, December 2013
Index of content:

Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, calculation of transport moments reduces to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders further than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders.
 ARTICLES


Partial Differential Equations

Multibump solutions for quasilinear elliptic equations with critical growth
View Description Hide DescriptionThe current paper is concerned with constructing multibump solutions for a class of quasilinear Schrödinger equations with critical growth. This extends the classical results of Coti Zelati and Rabinowitz [Commun. Pure Appl. Math.45, 1217–1269 (1992)] for semilinear equations as well as recent work of Liu, Wang, and Guo [J. Funct. Anal.262, 4040–4102 (2012)] for quasilinear problems with subcritical growth. The periodicity of the potentials is used to glue ground state solutions to construct multibump bound state solutions.

On fractional Schrdinger equation in with critical growth
View Description Hide DescriptionIn this paper, we study the following nonlinear fractional Schr dinger equation with critical exponent , where h is a small positive parameter, 0 < α < 1, , is the critical Sobolev exponent, and N > 2α, λ > 0 is a parameter. The potential is a positive continuous function satisfying some natural assumptions. By using variational methods, we obtain the existence of solutions in the following case: if , there exists λ0 > 0 such that for all λ ⩾ λ0, we show that it has one nontrivial solution and there exist at least nontrivial solutions; if , then there is one nontrivial solution and there exist at least nontrivial solutions for all λ > 0.

Existence of global weak solution for a reduced gravity two and a half layer model
View Description Hide DescriptionWe investigate the existence of global weak solution to a reduced gravity two and a half layer model in onedimensional bounded spatial domain or periodic domain. Also, we show that any possible vacuum state has to vanish within finite time, then the weak solution becomes a unique strong one.

Groupinvariant solutions of semilinear Schrödinger equations in multidimensions
View Description Hide DescriptionSymmetry group methods are applied to obtain all explicit groupinvariant radial solutions to a class of semilinear Schrödinger equations in dimensions n ≠ 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudoconformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schrödinger equations to groupinvariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of onedimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether's theorem, all conservation laws arising from these point symmetry groups are listed. Some groupinvariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative twodimensional form of the Schrödinger equations involving an extra modulation term with a parameter m = 2−n ≠ 0 is discussed.

Geometric solitons of Hamiltonian flows on manifolds
View Description Hide DescriptionIt is wellknown that the LIE (Locally Induction Equation) admit solitontype solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of isometries of the domain and the target space, respectively. With this insight, we propose the new concept of geometric solitons of Hamiltonian flows on manifolds, such as geometric Schrödinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schrödinger flow and geometric KdV flow, including magnetic curves as geometric Schrödinger solitons and explicit geometric KdV solitons on surfaces of revolution.

Nonexistence of global solution to ChernSimonsHiggs system
View Description Hide DescriptionIn this paper, we show that for a class of Higgs potentials V, the 2+1dimensional ChernSimonsHiggs system with negative energy or zero energy together with blows up in finite time.

On the essential spectrum of certain noncommutative oscillators
View Description Hide DescriptionWe show here that the spectrum of the family of noncommutative harmonic oscillators for in the range αβ = 1 is [0, +∞) and is entirely essential spectrum. The previous existing results concern the case αβ > 1 (case in which is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied), and ours therefore extend the picture to the range of parameters αβ ⩾ 1.

Infinitely many solutions for the nonlinear Schrödinger equations with magnetic potentials in
View Description Hide DescriptionIn this paper, we study a nonlinear Schrödinger equations with magnetic potentials in involving subcritical growth. Under some decaying and weak symmetry conditions of both electric and magnetic potentials, we prove that the equation has infinitely many nonradial complexvalued solutions by applying the finite reduction method.

L ^{2}stability of the VlasovMaxwellBoltzmann system near global Maxwellians
View Description Hide DescriptionWe present a L ^{2}stability theory of the VlasovMaxwellBoltzmann system for the twospecies collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L ^{2}distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.Y. and Xiao, Q.H., “A revisiting to the L ^{2}stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.Y., Yang, X.F., and Yun, S.B., “L ^{2} stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal.197, 657–688 (2010)] on the L ^{2}stability of the Boltzmann equation to the Boltzmann equation coupled with selfconsistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal largetime behavior of the VlasovMaxwellBoltzmann system in the whole space,” Commun. Pure Appl. Math.24, 1497–1546 (2011)] and Guo [“The VlasovMaxwellBoltzmann system near Maxwellians,” Invent. Math.153(3), 593–630 (2003)] satisfy a uniform L ^{2}stability estimate. This is the first result on the L ^{2}stability of the Boltzmann equation coupled with selfconsistent field equations in three dimensions.

Existence of multibump solutions for a class of Kirchhoff type problems in
View Description Hide DescriptionUsing variational methods, we establish existence of multibump solutions for a class of Kirchhoff type problems , where f is a continuous function with subcritical growth, V(x) is a critical frequency in the sense that . We show that if the zero set of V(x) has several isolated connected components Ω1, …, Ω k such that the interior of Ω i is not empty and ∂Ω i is smooth, then for λ > 0 large there exists, for any nonempty subset J ⊂ {1, …, k}, a bump solution is trapped in a neighborhood of ∪ j ∈ J Ω j .

Minimal blowup solutions of masscritical inhomogeneous Hartree equation
View Description Hide DescriptionIn this paper, we are concerned with the Cauchy problem of the inhomogeneous Hartree equation: , , N ⩾ 3. First, we establish the mass concentration property of the blowup solutions. Second, we show that the blowup solutions with minimal mass should concentrate at a critical point of k. Finally, under certain assumptions on global maximum points of k we establish nonexistence of such solutions.

Representation Theory and Algebraic Methods

Polytope expansion of Lie characters and applications
View Description Hide DescriptionThe weight systems of finitedimensional representations of complex, simple Lie algebras exhibit patterns beyond Weylgroup symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way. Since lattice polytopes are relatively simple, this decomposition is useful, in addition to being more economical than the decomposition into single weights. An expansion of characters into polytope sums follows from the polytope decomposition of weight systems. We study this polytope expansion here. A new, general formula is given for the polytope sums involved. The combinatorics of the polytope expansion are analyzed; we point out that they are reduced from those of the Weyl character formula (described by the Kostant partition function) in an optimal way. We also show that the weight multiplicities can be found easily from the polytope multiplicities, indicating explicitly the equivalence of the two descriptions. Finally, we demonstrate the utility of the polytope expansion by showing how polytope multiplicities can be used in the calculation of tensor product decompositions, and subalgebra branching rules.

Twisted vertex algebras, bicharacter construction and bosonfermion correspondences
View Description Hide DescriptionThe bosonfermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new bosonfermion correspondence of type DA. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and DA are isomorphisms of twisted vertex algebras.

Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
View Description Hide DescriptionWe present the first complete derivation of the wellknown asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol.

Lusztig symmetries and PoincareBirkhoffWitt basis for
View Description Hide DescriptionWe investigate a new kind of twoparameter weak quantized superalgebra , which is a weak Hopf superalgebra. It has a homomorphic image which is isomorphic to the usual twoparameter quantum superalgebra of . We also discuss the basis of by Lusztig's symmetries.

ManyBody and Condensed Matter Physics

Ward identities and chiral anomalies for coupled fermionic chains
View Description Hide DescriptionCoupled fermionic chains are usually described by an effective model written in terms of bonding and antibonding fermionic fields with linear dispersion in the vicinities of the respective Fermi points. We derive for the first time exact Ward Identities (WI) for this model, proving the existence of chiral anomalies which verify the AdlerBardeen nonrenormalization property. Such WI are expected to play a crucial role in the understanding of the thermodynamic properties of the system. Our results are nonperturbative and are obtained analyzing Grassmann functional integrals by means of constructive quantum field theory methods.

Aspects of the inverse problem for the Toda chain
View Description Hide DescriptionWe generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the subconstituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.

Quantum Mechanics

Absolutely continuous spectrum of the Schrödinger operator with a potential representable as a sum of three functions with special properties
View Description Hide DescriptionWe consider a family of Schrödinger operators depending on a real parameter and prove that the absolutely continuous spectrum covers the positive halfline for almost every value of the parameter. The potentials in this model are representable as a sum of three terms. This sum is neither fast decaying, nor it rapidly changes its sign or has derivatives with respect to the angular variables.

Application of polynomial su(1, 1) algebra to PöschlTeller potentials
View Description Hide DescriptionTwo novel polynomial su(1, 1) algebras for the physical systems with the first and second PöschlTeller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraic approach explores an appropriate new pair of raising and lowing operators of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derived naturally from the polynomial su(1, 1) algebras built by us.

Extending Romanovski polynomials in quantum mechanics
View Description Hide DescriptionSome extensions of the (thirdclass) Romanovski polynomials (also called Romanovski/pseudoJacobi polynomials), which appear in boundstate wavefunctions of rationally extended Scarf II and RosenMorse I potentials, are considered. For the former potentials, the generalized polynomials satisfy a finite orthogonality relation, while for the latter an infinite set of relations among polynomials with degreedependent parameters is obtained. Both types of relations are counterparts of those known for conventional polynomials. In the absence of any direct information on the zeros of the Romanovski polynomials present in denominators, the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of secondorder differential equations of Schrödinger type. It is also shown that on going from Scarf I to Scarf II or from RosenMorse II to RosenMorse I potentials, the variety of rational extensions is narrowed down from types I, II, and III to type III only.
