Volume 56, Issue 6, June 2015
Index of content:

The recent singlephoton doubleslit experiment of Steinberg et al., based on a weak measurement method proposed by Wiseman, showed that, by encoding the photon’s transverse momentum behind the slits into its polarization state, the momentum profile can subsequently be measured on average, from a difference of the separated fringe intensities for the two circular polarization components. They then integrated the measured average velocity field, to obtain the average trajectories of the photons enroute to the detector array. In this paper, we propose a modification of their experiment, to demonstrate that the average particle velocities and trajectories change when the mode of detection changes. The proposed experiment replaces a single detector by a pair of detectors with a given spacing between them. The pair of detectors is configured so that it is impossible to distinguish which detector received the particle. The pair of detectors is then analogous to the simple pair of slits, in that it is impossible to distinguish which slit the particle passed through. To establish the paradoxical outcome of the modified experiment, the theory and explicit threedimensional formulas are developed for the bilocal probability and current densities, and for the average velocity field and trajectories as the particle wavefunction propagates in the volume of space behind the Gaussian slits. Examples of these predicted results are plotted. Implementation details of the proposed experiment are discussed.
 ARTICLES

 Partial Differential Equations

A note on local behavior of eigenfunctions of the Schrödinger operator
View Description Hide DescriptionWe show that a real eigenfunction of the Schrödinger operator changes sign near some point in ℝ^{ n } under a suitable assumption on the potential.

Remarks on damped fractional Schrödinger equation with pure power nonlinearity
View Description Hide DescriptionWe investigate the initial value problem for a semilinear fractional damped Schrödinger equation. Global existence and scattering are proved depending on the size of the damping coefficient.

Constraint minimizers of mass critical Hartree energy functionals: Existence and mass concentration
View Description Hide DescriptionWe consider L ^{2}constraint minimizers of mass critical Hartree energy functionals in ℝ^{ N } with N ≥ 3. We prove that minimizers exist if and only if the parameter a > 0 satisfies , where Q is a positive radially symmetric ground state of in ℝ^{ N }. The blowup behavior of minimizers as a approaches a ^{∗} is also analyzed, for which all the mass concentrates at a global minimum point x 0 of the external potential V(x).

Interaction of jumpfan composite waves in a twodimensional jet for van der Waals gases
View Description Hide DescriptionWe consider a twodimensional (2D) jet by van der Waals gas streaming in parallel supersonic flow out of a duct into the atmosphere. We assume that the pressure p 0 of the oncoming uniform parallel flow is greater than the atmospheric pressure pA and belongs to . Then at the corners at exit the oncoming flow expands in two symmetric jumpfan (jf) composite waves to the atmospheric pressure. These two jf composite waves interact and emerge as simple waves from their zone of penetration. We present a mathematical analysis of the interaction of the jf composite waves. To construct the flow in the interaction region, we consider a discontinuous Goursat problem for the 2D isentropic irrotational steady Euler equations. The existence of global piecewise C ^{1} solution to the discontinuous Goursat problem is obtained constructively.

On the behavior of KazhikovSmagulov mass diffusion model for vanishing diffusion and viscosity coefficients
View Description Hide DescriptionWe consider the motion of a viscous incompressible fluid consisting of two components with a diffusion effect obeying Fick’s law in ℝ^{3}. We prove that there exists a small time interval where the fluid variables converge uniformly as the viscosity and the diffusion coefficient tend to zero. In the limit, we find a nonhomogeneous, nonviscous, incompressible fluid governed by an Eulerlike system.

Weak asymptotic methods for 3D selfgravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear EulerPoisson equations
View Description Hide DescriptionWe construct a family of classical continuous functions S(x, y, z, t, ϵ) which tend to satisfy asymptotically the system of selfgravitating pressureless fluids when ϵ → 0. This produces a weak asymptotic method in the sense of Danilov, Omel’yanov, and Shelkovich. The construction is based on a family of two ordinary differential equations (ODEs) (one for the continuity equation and one for the Euler equation) in classical Banach spaces of continuous functions. This construction applies to 3D selfgravitating pressureless fluids even in presence of point and string concentrations of matter. The method is constructive which permits to check numerically from standard methods for ODEs that these functions tend to the known or admitted solutions when the latter exist. As a direct application, we present a simulation of formation and evolution of a planetary system from a rotating disk of dust: a theorem in this paper asserts that the observed results are a depiction of functions that satisfy the system with arbitrary precision.

A degeneration of twophase solutions of the focusing nonlinear Schrödinger equation via RiemannHilbert problems
View Description Hide DescriptionTwophase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two and expressed in terms of the corresponding thetafunction. We show here that in a certain limiting regime, such solutions reduce to some elementary ones called “Solitons on unstable condensate.” This degeneration turns out to be conveniently studied by means of basic tools from the theory of RiemannHilbert problems. In particular, no acquaintance with Riemann surfaces and thetafunction is required for such analysis.
 Representation Theory and Algebraic Methods

Translation invariant pure state on and its split property
View Description Hide DescriptionIn this paper, we prove that a real lattice symmetric reflection positive translationinvariant pure state of admits split property, if and only if its twopoint spatial correlation functions decay exponentially. We use amalgamated representation of Cuntz algebras to represent twopoint spatial correlation functions on an augmented Hilbert space. The underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying twopoint correlation functions properties as spectral properties of a contractive selfadjoint operator on the augmented Hilbert space. Haag duality property of the pure state is crucially used in the analysis.

Homstructures on finitedimensional simple Lie superalgebras
View Description Hide DescriptionA Homstructure on a Lie superalgebra is an even linear mapping which twists the super Jacobi identity. In this paper, using Kac’s classification theorem and a reduction method, we show that finitedimensional simple Lie superalgebras over the complex field ℂ admit only the trivial Homstructures, that is, the scalar mappings.

Braids as a representation space of SU(5)
View Description Hide DescriptionThe standard model of particle physics provides very accurate predictions of phenomena occurring at the subatomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary is still unknown. Along the lines of previous preon models positing a substructure to explain these aspects, BilsonThompson showed how the first family of elementary particles is realized as the crossings of braids made of three strands, with charges resulting from twists of those strands with certain conditions; in this topological model, there are only two distinct neutrino states. Modeling the particles as braids implies these braids must be the representation space of a Lie algebra, giving the symmetries of the standard model. In this paper, this representation is made explicit, obtaining the raising operators associated with the Lie algebra of SU (5), one of the earliest grand unified theories. Because the braids form a group, the action of these operators are braids themselves, leading to their identification as gauge bosons. Possible choices for the other two families are also given. Although this realization of particles as braids is lacking a dynamical framework, it is very suggestive, especially when considered as a natural method of adding matter to loop quantum gravity.
 ManyBody and Condensed Matter Physics

Local perturbations perturb—exponentially–locally
View Description Hide DescriptionWe elaborate on the principle that for gapped quantum spin systems with local interaction, “local perturbations [in the Hamiltonian] perturb locally [the groundstate].” This principle was established by Bachmann et al. [Commun. Math. Phys. 309, 835–871 (2012)], relying on the “spectral flow technique” or “quasiadiabatic continuation” [M. B. Hastings, Phys. Rev. B 69, 104431 (2004)] to obtain locality estimates with subexponential decay in the distance to the spatial support of the perturbation. We use ideas of Hamza et al. [J. Math. Phys. 50, 095213 (2009)] to obtain similarly a transformation between gapped eigenvectors and their perturbations that is local with exponential decay. This allows to improve locality bounds on the effect of perturbations on the low lying states in certain gapped models with a unique “bulk ground state” or “topological quantum order.” We also give some estimate on the exponential decay of correlations in models with impurities where some relevant correlations decay faster than one would naively infer from the global gap of the system, as one also expects in disordered systems with a localized groundstate.

Gapped and gapless phases of frustrationfree spin chains
View Description Hide DescriptionWe consider a family of translationinvariant quantum spin chains with nearestneighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary twoqubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn (ψ) which is defined as the sum of rank1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn (ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal nonzero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustrationfree translationinvariant spin1/2 chains with nearestneighbor interactions.
 Quantum Mechanics

Analytical results of zerogap states in periodic potentials
View Description Hide DescriptionWe develop a method to construct various classes of onedimensional periodic potentials with two intersecting energy bands. Analytical exact results for the zerogap states are presented in an explicit form under certain parameter conditions. The position of the energies of these zerogap states in the energy bands is identified numerically.

Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions
View Description Hide DescriptionWe extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the twoparameter Lissajous systems on the sphere. These new families of superintegrable systems with integrals of arbitrary order are connected with Jacobi exceptional orthogonal polynomials of type I (or II) and supersymmetric quantum mechanics. Moreover, we present an algebraic derivation of the degenerate energy spectrum for the one and twoparameter Lissajous systems and the rationally extended models. These results are based on finitely generated polynomial algebras, Casimir operators, realizations as deformed oscillator algebras, and finitedimensional unitary representations. Such results have only been established so far for 2D superintegrable systems separable in Cartesian coordinates, which are related to a class of polynomial algebras that display a simpler structure. We also point out how the structure function of these deformed oscillator algebras is directly related with the generalized Heisenberg algebras spanned by the nonpolynomial integrals.

Rational extensions of the trigonometric DarbouxPöschlTeller potential based on paraJacobi polynomials
View Description Hide DescriptionThe possibility for the Jacobi equation to admit, in some cases, general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them paraJacobi polynomials. Such polynomials are used here to build seed functions of a DarbouxBäcklund transformation for the trigonometric DarbouxPöschlTeller potential. As a result, onestep regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter λ are constructed. For each n value, the eigenstates of these extended potentials are associated with a novel family of λdependent polynomials, which are orthogonal on .

Geometry of Winter model
View Description Hide DescriptionBy constructing the Riemann surface controlling the resonance structure of Winter model, we determine the limitations of perturbation theory. We then derive explicit nonperturbative results for various observables in the weakcoupling regime, in which the model has an infinite tower of longlived resonant states. The problem of constructing proper initial wavefunctions coupled to single excitations of the model is also treated within perturbative and nonperturbative methods.

Analytic structure of the Smatrix for singular quantum mechanics
View Description Hide DescriptionThe analytic structure of the Smatrix of singular quantum mechanics is examined within a multichannel framework, with primary focus on its dependence with respect to a parameter (Ω) that determines the boundary conditions. Specifically, a characterization is given in terms of salient mathematical and physical properties governing its behavior. These properties involve unitarity and associated currentconserving Wronskian relations, timereversal invariance, and Blaschke factorization. The approach leads to an interpretation of effective nonunitary solutions in singular quantum mechanics and their determination from the unitary family.

Bilocal current densities and mean trajectories in a Young interferometer with two Gaussian slits and two detectors
View Description Hide DescriptionThe recent singlephoton doubleslit experiment of Steinberg et al., based on a weak measurement method proposed by Wiseman, showed that, by encoding the photon’s transverse momentum behind the slits into its polarization state, the momentum profile can subsequently be measured on average, from a difference of the separated fringe intensities for the two circular polarization components. They then integrated the measured average velocity field, to obtain the average trajectories of the photons enroute to the detector array. In this paper, we propose a modification of their experiment, to demonstrate that the average particle velocities and trajectories change when the mode of detection changes. The proposed experiment replaces a single detector by a pair of detectors with a given spacing between them. The pair of detectors is configured so that it is impossible to distinguish which detector received the particle. The pair of detectors is then analogous to the simple pair of slits, in that it is impossible to distinguish which slit the particle passed through. To establish the paradoxical outcome of the modified experiment, the theory and explicit threedimensional formulas are developed for the bilocal probability and current densities, and for the average velocity field and trajectories as the particle wavefunction propagates in the volume of space behind the Gaussian slits. Examples of these predicted results are plotted. Implementation details of the proposed experiment are discussed.

Quantum aspects of a moving magnetic quadrupole moment interacting with an electric field
View Description Hide DescriptionThe quantum dynamics of a moving particle with a magnetic quadrupole moment that interacts with electric and magnetic fields is introduced. By dealing with the interaction between an electric field and the magnetic quadrupole moment, it is shown that an analogue of the Coulomb potential can be generated and bound state solutions can be obtained. Besides, the influence of the Coulombtype potential on the harmonic oscillator is investigated, where bound state solutions to both repulsive and attractive Coulombtype potentials are achieved and the arising of a quantum effect characterized by the dependence of the harmonic oscillator frequency on the quantum numbers of the system is discussed.

Coherent states for nonlinear harmonic oscillator and some of its properties
View Description Hide DescriptionA onedimensional nonlinear harmonic oscillator is studied in the context of generalized coherent states. We develop a perturbative framework to compute the eigenvalues and eigenstates for the quantum nonlinear oscillator and construct the generalized coherent states based on GazeauKlauder formalism. We analyze their statistical properties by means of Mandel parameter and second order correlation function. Our analysis reveals that the constructed coherent states exhibit superPoissonian statistics. Moreover, it is shown that the coherent states mimic the phenomena of quantum revivals and fractional revivals during their time evolution. The validity of our results has been discussed in terms of various parametric bounds imposed by our computational scheme.