Volume 55, Issue 3, March 2014
Index of content:

The inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the socalled theta condition, and the formulation of the inverse problem in terms of a RiemannHilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.
 ARTICLES


Partial Differential Equations

A survey of solutions in a gravitational BornInfeld theory
View Description Hide DescriptionAn elliptic equation that arises from a cosmic string model with the action of the BornInfeld nonlinear electromagnetism, is considered. We classify and establish the uniqueness of radially symmetric solutions.

On a regularization of a scalar conservation law with discontinuous coefficients
View Description Hide DescriptionThis paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients. It is clear that the delta standing wave should be introduced into the Riemann solution in some nonclassical situation. In order to study the formation of delta standing wave, we consider a regularization of the discontinuous coefficient with the Helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter ɛ > 0. The regularization mechanism is a nonlinear bending of characteristic curves that prevents their finitetime intersection. It is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the ɛ → 0 limit. Compared with the classical method of vanishing viscosity, here it is clear to see how the delta standing wave forms naturally along the characteristics.

Wave kernels for the Dirac, Euler operators and the harmonic oscillator
View Description Hide DescriptionExplicit solutions for the wave equations associated to the Dirac, Euler operators and the harmonic oscillator are given.

Continuous properties of the solution map for the Euler equations
View Description Hide DescriptionIn this paper, we study the dependence on initial data of solutions to the incompressible Euler equations in Besov spaces. We show that for s > n/p + 1, p ∈ (1, ∞), and r ∈ [1, ∞], the solution map u 0↦u is Hölder continuous in Besov space equipped with weaker topology. When the space variable x is taken to be periodic, we obtain a family of explicit periodic solutions. Furthermore, we prove that for any and 1 ⩽ r ⩽ ∞, the solution map is not globally uniformly continuous in , which extends some results of Himonas and Misiołek [Commun. Math. Phys.296(1), 285–301 (2010)].

Remarks on the regularity criteria of threedimensional magnetohydrodynamics system in terms of two velocity field components
View Description Hide DescriptionWe study the threedimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely. The proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field. This result may be seen as a component reduction result of many previous works [C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations,” J. Differ. Equ.213(2), 234–254 (2005); Y. Zhou, “Remarks on regularities for the 3D MHD equations,” Discrete Contin. Dyn. Syst.12(5), 881–886 (2005)].

Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions
View Description Hide DescriptionThe inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the socalled theta condition, and the formulation of the inverse problem in terms of a RiemannHilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.

The existence and concentration of positive solutions for a nonlinear SchrödingerPoisson system with critical growth
View Description Hide DescriptionWe consider the SchrödingerPoisson system: −ε^{2}Δu + V(x)u + ϕ(x)u = f(u),−Δϕ = u ^{2} in , where the nonlinear term f is of critical growth. In this paper, we construct a solution (u ɛ, ϕɛ) of the above elliptic system, which concentrates at an isolated component of positive locally minimum points of V as ɛ → 0 under certain conditions on f. In particular, the monotonicity of and the socalled AmbrosettiRabinowitzcondition are not required.

Multipeak solution for nonlinear magnetic Choquard type equation
View Description Hide DescriptionIn this paper, we study a class of nonlinear magnetic Choquard type equation involving a magnetic potential and nonlocal nonlinearities. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential V.

Representation Theory and Algebraic Methods

Radford's biproducts and YetterDrinfeld modules for monoidal HomHopf algebras
View Description Hide DescriptionLet (H, α) be a monoidal Hombialgebra and (B, β) be a left (H, α)Hommodule algebra and also a left (H, α)Homcomodule coalgebra. Then in this paper, we first introduce the notion of a Homsmash coproduct, which is a monoidal Homcoalgebra. Second, we find sufficient and necessary conditions for the Homsmash product algebra structure and the Homsmash coproduct coalgebra structure on B ⊗ H to afford B ⊗ H a monoidal Hombialgebra structure, generalizing the wellknown Radford's biproduct, where the conditions are equivalent to that (B, β) is a bialgebra in the category of HomYetterDrinfeld modules . Finally, we illustrate the category of HomYetterDrinfeld modules and prove that the category is a braided monoidal category.

The category of YetterDrinfel'd Hommodules and the quantum HomYangBaxter equation
View Description Hide DescriptionIn this paper, we introduce the category of YetterDrinfel'd Hommodules which is a braided monoidal category and show that the category of YetterDrinfel'd Hommodules is a full monoidal subcategory of the left center of left Hommodule category. Also we study the equivalent relationship between the category of YetterDrinfel'd Hommodules and the category of Hommodules over the Drinfel'd double. Finally, the FaddeevReshetikhinTakhtajan (FRT) type theorem for the quantum HomYangBaxter equation is investigated.

Quantum Mechanics

Mappings of open quantum systems onto chain representations and Markovian embeddings
View Description Hide DescriptionWe study systems coupled linearly to a bath of oscillators. In an iterative process, the bath is transformed into a chain of oscillators with nearest neighbour interactions. A systematic procedure is provided to obtain the spectral density of the residual bath in each step, and it is shown that under general conditions these data converge. That is, the asymptotic part of the chain is universal, translation invariant with semicircular spectral density. The methods are based on orthogonal polynomials, in which we also solve the outstanding socalled “sequence of secondary measures problem” and give them a physical interpretation.

Scattering problems in the fractional quantum mechanics governed by the 2D spacefractional Schrödinger equation
View Description Hide DescriptionThe 2D spacefractional Schrödinger equation in the timeindependent and timedependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.

Smallenergy analysis for the selfadjoint matrix Schrödinger operator on the half line. II
View Description Hide DescriptionThe matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the smallenergy asymptotics for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.

A parametric approach to supersymmetric quantum mechanics in the solution of Schrödinger equation
View Description Hide DescriptionWe study exact solutions of the Schrödinger equation for some potentials. We introduce a parametric approach to supersymmetric quantum mechanics to calculate energy eigenvalues and corresponding wave functions exactly. As an application we solve Schrödinger equation for the generalized Morse potential, modified Hulthen potential, deformed RosenMorse potential and PoschlTeller potential. The method is simple and effective to get the results.

Oscillators in a (2+1)dimensional noncommutative space
View Description Hide DescriptionWe study the Harmonic and Dirac Oscillator problem extended to a threedimensional noncommutative space where the noncommutativity is induced by the shift of the dynamical variables with generators of in a unitary irreducible representation considered in Falomir et al. [Phys. Rev. D86, 105035 (2012)]. This redefinition is interpreted in the framework of the Levi's decomposition of the deformed algebra satisfied by the noncommutative variables. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncommutativity parameters. Since the representation space of the unitary irreducible representations can be realized in terms of spaces of squareintegrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

Scattering theory with path integrals
View Description Hide DescriptionStarting from wellknown expressions for the Tmatrix and its derivative in standard nonrelativistic potential scattering, I rederive recent pathintegral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.

Quantum Information and Computation

Process tomography for unitary quantum channels
View Description Hide DescriptionWe study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a dlevel quantum system can be uniquely identified among all other channels (unitary or otherwise) with only O(d ^{2}) interactive observables, as opposed to the O(d ^{4}) required for tomography of arbitrary channels. This result generalizes to the problem of identifying channels with at most q Kraus operators, and slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with q Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.

Structure and properties of the algebra of partially transposed permutation operators
View Description Hide DescriptionWe consider the structure of algebra of operators, acting in nfold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semisimple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreducible representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1).

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

The finite and largeN behaviors of independentvalue matrix models
View Description Hide DescriptionWe investigate the finite and large N behaviors of independentvalue O(N)invariant matrix models. These are models defined with matrixtype fields and with no gradient term in their action. They are generically nonrenormalizable but can be handled by nonperturbative techniques. We find that the functional integral of any O(N) matrix trace invariant may be expressed in terms of an O(N)invariant measure. Based on this result, we prove that, in the limit that all interaction coupling constants go to zero, any interacting theory is continuously connected to a pseudofree theory. This theory differs radically from the familiar free theory consisting in putting the coupling constants to zero in the initial action. The proof is given for generic, finitesize matrix models, whereas, in the limiting case N → ∞, we succeed in showing this behavior for restricted types of actions using a particular scaling of the parameters.

Multitime Schrödinger equations cannot contain interaction potentials
View Description Hide DescriptionMultitime wave functions are wave functions that have a time variable for every particle, such as . They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for noninteracting particles, it is a challenge to set up consistent multitime equations with interaction. We prove for a wide class of multitime Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multitime wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cutoff length δ > 0 is introduced (in the sense that the multitime wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multitime Schrödinger equations with interaction potentials of range δ are consistent; however, in the desired limit δ → 0 of removing the cutoff, the resulting multitime equations are interactionfree, which supports the conclusion expressed in the title.
