A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

Spinor with Schrödinger symmetry and non-relativistic supersymmetry

We construct the d-dimensional “half” Schrödinger equation, which is a kind of the root of the Schrödinger equation, from the (d+1)-dimensional free Dirac equation. The solution of the “half” Schrödinger equation also satisfies the usual free Schrödinger equation. We also find that the explicit transformation laws of the Schrödinger and the half Schrödinger fields under the Schrödinger symmetry transformation are derived by starting from the Klein-Gordon equation and the Dirac equation in d+1 dimensions. We derive the 3- and 4-dimensional super-Schrödinger algebra from the superconformal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one (complex) spinor fields and the explicit transformation properties have been found.     

Phase diagram of the Gross-Neveu model: exact results and condensed matter precursors

Recently the revised phase diagram of the (large N) Gross-Neveu model in 1 + 1 dimensions with discrete chiral symmetry has been determined numerically. It features three phases, a massless and a massive Fermi gas and a kink-antikink crystal. Here we investigate the phase diagram by analytical means, mapping the Dirac-Hartree-Fock equation onto the non-relativistic Schrödinger equation with the (single gap) Lamé potential. It is pointed out that mathematically identical phase diagrams appeared in the condensed matter literature some time ago in the context of the Peierls-Fröhlich model and ferromagnetic superconductors.     

Low energy asymptotics for Schrödinger operators with slowly decreasing potentials

Low energy behavior of Schrödinger operators with potentials which decay slowly at infinity is studied. It is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0. This implies rapid local decay for the solutions of the Schrödinger equation. On the other hand, if the potential is negative then the resolvent has discontinuity at zero energy. Thus one cannot expect local decay faster than ordert ^–1 ast.     

Analysis on generation schemes of Schrödinger cat-like states under experimental imperfections

An analysis and a comparison of two generation schemes of Schrödinger cat-like state including experimental imperfections are presented. Under practical conditions, a scheme using a squeezed vacuum and a photon subtraction will generate a cat-like state with its fidelity to the Schrödinger cat state F = 0.815 and value of its Wigner function at the origin of the phase space W(0,0) = −0.203, and then turned out to be more feasible than the scheme using squeezed single-photon state. The non-classicality of these cat-like states is governed only by non-classical photon number statistics. The criteria for ensuring W(0,0) < 0 are also presented in terms of imperfection parameter diagrams.     

First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions

The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H, H₁) of intertwined Hamiltonians one can associate another pair of second-order partial differential operators (R, R₁), related to the same intertwining operator and such that H (resp. H₁) commutes with R (resp. R₁). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.     

Bose-Einstein solitons in time-dependent linear potential

In this paper, Bose-Einstein soliton solutions of the nonlinear Schrödinger equation with time-dependent linear potential are considered. Based on the F-expansion method, we present a number of Jacobian elliptic function solutions. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric functions, respectively. Specially, for V_ext = ZF(T) = Z[mg + Hcos (ω₁T)], we discussed the Bose-Einstein condensate trapped in the coupling external field with considering the effect of gravity; for F(T) = constant, it describes the wave (Langmuir or electromagnetic) in a linearly inhomogeneous plasma with cubic nonlinearly.     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

Exact, zero-energy, square-integrable solutions of a model related to the Maxwell’s fish-eye problem

A model, which admits normalizable wave functions of the Schrödinger equation at the energy of E = 0, is exactly solved and the solutions are compared to the corresponding classical trajectories. The wave functions are proved to be square-integrable for discrete (quantized) values of the coupling constant of the used potential. We also show that our model is a specific version of the well-known Maxwell’s fish-eye. This is performed with the help of a suitably chosen conformal mapping.     

Multipole surface plasmon resonance of an aluminium surface

The surface photoelectric effect and the surface plasmon resonances appear when a p/transverse magnetic polarized laser hits a gas-solid interface. We model this effect in the long wave length (LWL) domain (λ_vac > 10 nm, ?ω < 124 eV) by combining the Ampère-Maxwell equation, written in classical approximation, with the material equation for the susceptibility. The resulting model, called the vector potential from the electron density (VPED), calculates the susceptibility as a product of the bulk susceptibility and the electron density of the actual system. The bulk susceptibility is a sum of the bound electron scalar susceptibility taken from the experiment and of the conduction electron non-local isotropic susceptibility tensor in a jellium metal (Lindhard, 1954 [1]). The electron density is the square of the wave function solution of the Schrödinger equation. The analysis of observables, the reflectance R and the photoelectron yield Y as well as the induced charge density permits to identify and characterize the multipole surface plasmon resonance of Al(111) appearing at ω_m ∼ 0.8ω_p or 11-12 eV.     

Optical analogue of the Aharonov-Bohm effect using anisotropic media

We show that in the context of paraxial optics, which can be analyzed through a wave equation similar to the non-relativistic Schrödinger equation of quantum mechanics but replacing time t by spatial coordinate z, the existence of a vector potential A_⊥ mimicking the magnetic vector potential in quantum mechanics is allowed by specific gauge symmetries of the optical field in a medium with anisotropic refractive index. In this way, we use Feynman?s path integral to demonstrate an optical analogue of the quantum-mechanical Aharonov-Bohm effect, encouraging the search for another optical systems with analogies with more complex quantum field theories.     

Spatiotemporal self-similar waves for the (3 + 1)-dimensional inhomogeneous cubic-quintic nonlinear medium

Spatiotemporal self-similar waves of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation, describing propagation of optical pulses in a cubic-quintic nonlinear medium with inhomogeneous dispersion and gain, are derived. A one-to-one correspondence between such self-similar waves and solutions of the constant-coefficient cubic-quintic nonlinear Schrödinger equation exists when two certain compatibility conditions are satisfied. Under these conditions, we discuss dynamical behaviors of self-similar waves in dispersion decreasing fiber.     

Theoretical modeling for quantum-confined Stark effect due to internal piezoelectric fields in GaInN strained quantum wells

Recent experimental investigations revealed that the biaxial stress in thin InGaN layers grown on thick GaN layer induces a large piezoelectric field along [0001] orientation that causes red-shift in optical transitions and reduction in oscillator strengths because of spatial separation of the electron and hole wave functions. In this Letter based on theoretical modeling we determined the well width z-dependent effect on red-shifted quantum-confined Stark effect (QCSE) in GaN/In_xGa_1 − xN (x=0.13) strained quantum well structures. Analyses are based on the solution of Schrödinger equation in a finite well including the internal piezoelectric electric field (F) due to the strained polarization as the perturbation potential. Our theoretical results show: (1) the red-shift in optical transition has a quadratic well-width form as it is for infinite wells (Davies, 1998) [1], (2) assuming the model based on a carrier effective mass dependence on the width of quantum wells, m^∗(z), fits the experimental data (Takeuchi et al., 1997) [2] much more accurate compare to the model with constant effective mass, m^∗.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G^′/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before.     

The generalized nonlinear Schrödinger hierarchy, its integrable coupling system and their bi-Hamiltonian structure

Based on a subalgebra G of Lie algebra A₂, a new Lie algebra G^∗ is constructed. By making use of the Tu scheme, the generalized nonlinear Schrödinger hierarchy and its integrable coupling are both obtained with the help of their corresponding special loop algebras. At last, by means of the quadratic-form identity, their bi-Hamiltonian structures of the generalized nonlinear Schrödinger hierarchy and its integrable coupling system are worked out respectively. The approach presented in this Letter can be used in other integrable hierarchies.     

On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form , where the real field ?(t,xi) is some solution of nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field ?. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.     

Appearance of a purely singular continuous spectrum in a class of random schrödinger operators

We consider a discrete Schrödinger operator on l²() with a random potential decaying at infinity as ¦n¦^–1/2. We prove that its spectrum is purely singular. Together with previous results, this provides simple examples of random Schrödinger operators having a singular continuous component in its spectrum.     

Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields _t (x) and _t (x) which diagonalizes the field-HamiltonianH _t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's.The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent.Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.     

High energy femtosecond supercontinuum light generation in large mode area photonic crystal fiber

A large mode area photonic crystal fiber (LMA PCF) with an effective area of 180 μm² is used to generate a high energy, micro-joule range, flat, octave spanning supercontinuum (SC) extending from ~ 600 nm to ~ 1720 nm. A train of femtosecond pulses from a widely-tunable parametric amplifier pumped by a Ti:Sapphire regenerative amplifier system are coupled into a 20 cm length of LMA PCF generating a SC of 1.4 μJ energy. We present an experimental study of the high energy SC as a function of the input power and the pumping wavelength. The spectrum obtained at a pump wavelength of 1260 nm presents spectral flatness variation less than 12 dB over more than 1.1 octave bandwidth. The physical processes behind the SC formation are described in the normal and the anomalous dispersion regions. Our experimental results are successfully compared with the numerical solution of the nonlinear Schrödinger equation.