Volume 54, Issue 4, April 2013
Index of content:

We study a model Schrödinger operator with constant magnetic field on an infinite wedge with Neumann boundary condition. The magnetic field is assumed to be tangent to a face. We compare the bottom of the spectrum to the model spectral quantities coming from the regular case. We are particularly motivated by the influence of the magnetic field and the opening angle of the wedge on the spectrum of the model operator and we exhibit cases where the bottom of the spectrum is smaller than in the regular case. Numerical computations enlighten the theoretical approach.
 ARTICLES


Partial Differential Equations

Similarity solutions of FokkerPlanck equation with moving boundaries
View Description Hide DescriptionIn this work we present new exact similarity solutions with moving boundaries of the FokkerPlanck equation having both timedependent drift and diffusion coefficients.

Global solvability and blow up for the convective CahnHilliard equations with concave potentials
View Description Hide DescriptionWe study initial boundary value problems for the unstable convective CahnHilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is wellknown that without the convective term, the solutions of the unstable CH equation may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂_{ x } u in the CahnHilliard equation prevents blow up at least for . We also show that the blowing up solutions still exist if p is large enough (p ⩾ 2). The related equations like KolmogorovSivashinskySpiegel equation, sixth order convective CahnHilliard equation, are also considered.

On the fast diffusion with strong absorption
View Description Hide DescriptionIn the present paper the initial boundary value problem for the fast diffusion equation with strong absorption is considered. An optimal condition guaranteeing the strict positivity of the solution is proposed.

Selfadjoint extensions of Dirac operators with Coulomb type singularity
View Description Hide DescriptionIn this work we construct selfadjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a HardyDirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.

Nonlinear Schroedinger equation in the presence of uniform acceleration
View Description Hide DescriptionWe consider a recently proposed nonlinear Schroedinger equation exhibiting solitonlike solutions of the powerlaw form , involving the qexponential function which naturally emerges within nonextensive thermostatistics [ , with ]. Since these basic solutions behave like free particles, obeying p = ℏk, E = ℏω, and E = p ^{2}/2m (1 ⩽ q < 2), it is relevant to investigate how they change under the effect of uniform acceleration, thus providing the first steps towards the application of the aforementioned nonlinear equation to the study of physical scenarios beyond free particle dynamics. We investigate first the behaviour of the powerlaw solutions under Galilean transformation and discuss the ensuing Dopplerlike effects. We consider then constant acceleration, obtaining new solutions that can be equivalently regarded as describing a free particle viewed from an uniformly accelerated reference frame (with acceleration a) or a particle moving under a constant force −ma. The latter interpretation naturally leads to the evolution equation with V(x) = max. Remarkably enough, the potential V couples to Φ^{ q }, instead of coupling to Φ, as happens in the familiar linear case (q = 1).

Reduction of balance laws to conservation laws by means of equivalence transformations
View Description Hide DescriptionA class of partial differential equations (a conservation law and two balance laws), with three independent variables and involving six arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent, and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, we are interested in finding an equivalent autonomous system of conservation laws. It is shown how the results apply to some physical problems.

The Schrödinger operator on an infinite wedge with a tangent magnetic field
View Description Hide DescriptionWe study a model Schrödinger operator with constant magnetic field on an infinite wedge with Neumann boundary condition. The magnetic field is assumed to be tangent to a face. We compare the bottom of the spectrum to the model spectral quantities coming from the regular case. We are particularly motivated by the influence of the magnetic field and the opening angle of the wedge on the spectrum of the model operator and we exhibit cases where the bottom of the spectrum is smaller than in the regular case. Numerical computations enlighten the theoretical approach.

Representation Theory and Algebraic Methods

Weak Lie symmetry and extended Lie algebra^{a)}
View Description Hide DescriptionThe concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found (“extended Lie algebras”) which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended CartanKilling form. Transformation groups from nonrelativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1dimensional gravitational fields).

Quantum Mechanics

Phase space pathintegral formulation of the abovethreshold ionization
View Description Hide DescriptionAtoms and molecules submitted to a strong laser field can emit electrons of high energies in the abovethreshold ionization (ATI) process. This process finds a highly intuitive and also quantitative explanation in terms of Feynman's path integral and the concept of quantum orbits [P. Salières et al. , Science292, 902 (Year: 2001)]10.1126/science.108836. However, the connection with the Feynman pathintegral formalism is explained only by intuition and analogy and within the socalled strongfield approximation (SFA). Using the phase space pathintegral formalism we have obtained an exact result for the momentumspace matrix element of the total timeevolution operator. Applying this result to the ATI we show that the SFA and the socalled improved SFA are, respectively, the zeroth and the firstorder terms of the expansion in powers of the laserfree effective interaction of the electron with the rest of the atom (molecule). We have also presented the secondorder term of this expansion which is responsible for the ATI with double scattering of the ionized electron.

New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
View Description Hide DescriptionIn recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higherorder integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finitedimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

Inverse scattering for lasso graph
View Description Hide DescriptionThe inverse problem for the magnetic Schrödinger operator on the lasso graph with different matching conditions at the vertex is investigated. It is proven that the TitchmarshWeyl function known for different values of the magnetic flux through the cycle determines the unique potential on the loop, provided the entries of the vertex scattering matrix S parametrizing matching conditions satisfy s _{12} s _{23} s _{31} ≠ s _{13} s _{21} s _{32}. This is in contrast to numerous examples showing that the potential on the loop cannot be reconstructed from the boundary measurements.

Massive Dirac equation in asymmetric Hulthén potential
View Description Hide DescriptionOnedimensional effective mass Dirac equation is solved in asymmetric Hulthén potential. Scattering and bound state solutions are obtained in terms of hypergeometric functions. Transmission and reflection coefficients are calculated by matching conditions on the wave function. The necessary conditions for the existence of transmission resonances and supercriticality are derived.

Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum
View Description Hide DescriptionWe consider a system of N pairwise interacting particles described by the Hamiltonian H, where σ_{ ess }(H) = [0, ∞) and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if N ⩾ 4 and none of the Hamiltonians corresponding to the subsystems containing N − 2 or less particles has an eigenvalue equal to zero then H has a finite number of negative energy bound states. This result provides a positive proof to a longstanding conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the “true” fourbody Efimov effect.

Bound states and scattering coefficients of selfadjoint Hamiltonians with a mass jump
View Description Hide DescriptionPhysical selfadjoint extensions and their spectra of the simplest onedimensional Hamiltonian operator in which the mass is constant except for a finite jump at one point of the real axis are correctly found. Some selfadjoint extensions are used to model different kinds of semiconductor heterojunctions within the effectivemass approximation. Their properties and relation to different boundary conditions on envelope wave functions are studied. The limiting case of equal masses (with no mass jump) is reviewed.

Functional Wigner representation of quantum dynamics of BoseEinstein condensate
View Description Hide DescriptionWe develop a method of simulating the full quantum field dynamics of multimode multicomponent BoseEinstein condensates in a trap. We use the truncated Wigner representation to obtain a probabilistic theory that can be sampled. This method produces cnumber stochastic equations which may be solved using conventional stochastic methods. The technique is valid for large mode occupation numbers. We give a detailed derivation of methods of functional Wigner representation appropriate for quantum fields. Our approach describes spatial evolution of spinor components and properly accounts for nonlinear losses. Such techniques are applicable to calculating the leading quantum corrections, including effects such as quantum squeezing, entanglement, EPR correlations, and interactions with engineered nonlinear reservoirs. By using a consistent expansion in the inverse density, we are able to explain an inconsistency in the nonlinear loss equations found by earlier authors.

Spectral properties of a confined nonlinear quantum oscillator in one and three dimensions
View Description Hide DescriptionWe analyze the spectral behaviour of a nonlinear quantum oscillator model under confinement. The underlying potential is given by a harmonic oscillator interaction plus a nonlinear term that can be weakened or strengthened through a parameter. Numerical eigenvalues of the model in one and three dimensions are presented. The asymptotic behaviour of the eigenvalues for confinement relaxation and for vanishing nonlinear term in the potential is investigated. Our findings are compared with existing results.

Full counting statistics of stationary particle beams
View Description Hide DescriptionWe present a general theoretical framework for treating particle beams as timestationary limits of many particle systems. Due to stationarity, the total particle number diverges, and a description in Fock space is no longer possible. Nevertheless, we show that when describing the particle detection via second quantized arrival time observables, such beams exhibit a welldefined “local” counting statistics, that is, full counting statistics of all clicks falling into any given finite time interval. We also treat in detail a realization of such a beam via the long time limit of a source creating particles in a fixed initial state from which they then evolve freely. From the mathematical point of view, the beam is described by a quasifree state which, in the oneparticle level, is locally trace class with respect to the operator valued measure describing the time observable; this ensures the existence of a Fredholm determinant defining the characteristic function of the counting statistics.

An area law for the bipartite entanglement of disordered oscillator systems
View Description Hide DescriptionWe prove an upper bound proportional to the surface area for the bipartite entanglement of the ground state and thermal states of harmonic oscillator systems with disorder, as measured by the logarithmic negativity. Our assumptions are satisfied for some standard models that are almost surely gapless in the thermodynamic limit.

Quantum Information and Computation

Quantumtoclassical rate distortion coding
View Description Hide DescriptionWe establish a theory of quantumtoclassical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit a classical description of the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a singleletter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a nonclassical one, in which she performs a collective measurement on successive outputs of the source.

Partial transpose of random quantum states: Exact formulas and meanders
View Description Hide DescriptionWe investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.
