The equation X∇detX=detX∇trX,multiplication of cofactor pair systems,and the Levi-Civita equivalence problem

Cofactor pair systems generalize the separable potential Hamiltonian systems. They admit $n$ quadratic integrals of motion, they have a bi-Hamilton formulation, they are completely integrable and they are equivalent to separable Lagrangian systems. Cofactor pair systems can be constructed through a peculiar multiplicative structure of the so-called quasi-Cauchy–Riemann equations ${\left(\mathrm{cof}J\right)}^{-1}\nabla V={\left(\mathrm{cof}\tilde{J}\right)}^{-1}\nabla \tilde{V}$, where $J$ and $\tilde{J}$ are special conformal Killing tensors.In this work we have isolated the properties that are responsible for the multiplication, allowing us to give an elegant characterization of systems that admit multiplication. In this characterization the equation $X\nabla detX=detX\nabla \mathrm{tr}X$ plays a central role.We describe how multiplication of quasi-Cauchy–Riemann equations can be considered as a special case of a more general kind of multiplication, defined on the solution space of certain systems of partial differential equations. We investigate algebraic properties of this multiplication and provide several methods for constructing new systems with multiplicative structure. We also discuss the role of the multiplication in the theory of equivalent dynamical systems on Riemannian manifolds, developed by Levi-Civita.     

Geometrical equivalence of a deformed heisenberg spin equation and the generalized nonlinear schrödinger equation

It is shown that the general SO(3) invariant deformed Heisenberg spin chain discussed by Mikhailov and Shabat is geometrically equivalent to a generalized nonlinear Schrödinger equation through a moving space curve formalism. They are also mutually gauge equivalent in the sense noted by Kundu in a different context.     

The second hyperpolarizability of systems described by the space-fractional Schrödinger equation

The static second hyperpolarizability is derived from the space-fractional Schrödinger equation in the particle-centric view. The Thomas–Reiche–Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second hyperpolarizability for a space-fractional quantum system. The total oscillator strength is shown to decrease as the space-fractional parameter α decreases, which reduces the optical response of a quantum system in the presence of an external field. This damped response is caused by the wavefunction dependent position and momentum commutation relation. Although the maximum response is damped, we show that the one-dimensional quantum harmonic oscillator is no longer a linear system for $\alpha \ne 1$, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of $\alpha \to 1/2$.     

W-algebras and the equivalence of bihamiltonian,Drinfeld–Sokolov and Dirac reductions

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld–Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.     

Gauge equivalence,supersymmetry and classical solutions of the ospu (1, 1/1) Heisenberg model and the nonlinear Schrödinger equation

An integrable generalization of the continuous classical O(2, 1) pseudospin Heisenberg model to the case of the ospu(1, 1/1) superalgebra is constructed. The gauge equivalence of the constructed model and the related NLSE is established. We indicate a method of generating classical solutions using the global ospu(1, 1/1) supersymmetry. The relationship between solutions of O(2, 1) HM and superpartners of NLSE is obtained.     

Generalized nonlinear Doebner-Goldin Schrödinger equation and the relaxation of quantum systems

We propose a new class of nonlinear homogeneous extension of the Doebner-Goldin Schrödinger equation, valid for arbitrary representations and operators, chosen in accordance with the investigated physical problem. We verify that the nonlinearity simulates an environment, thence, the new model leads to simple exact solutions as, for instance, the time-dependent squeezed coherent states and a special class of stationary states that we call pseudothermal, reached after relaxation. We illustrate the use of the new equation with applications to problems such as, the relaxation of a two-level or spin-1/2 system, and of the harmonic oscillator (HO) or equivalently, the emission-absorption process of photons in an electromagnetic cavity. Furthermore, in order to compare solutions for the HO example we introduce two different representations in the new equation, one continuous (positional representation) and the other discrete (Fock states).     

Gauge equivalence and the inverse spectral problem for the magnetic Schrödinger operator on the torus

We study the inverse spectral problem for the Schrödinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V(x). Guillemin [11] proved that the spectrum of H determines B(x) and V(x). A simple proof of Guillemin’s results was given by the authors in [3]. In the present paper, we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that, generically, the spectrum and the magnetic field determine the “extended” gauge equivalence class of the magnetic potential. The proof is a modification of that in [3] with some corrections and clarifications.     

The problem with “The problem of the Pegg–Barnett phase operator”

We show that the physically absurd state obtained by Vorontsov and Rembovsky [Phys. Lett. A 254 (1999) 7] results from an unphysical choice of approximate phase measurement and not from the phase formalism itself as claimed. Thus their assertion that it is impossible to perform even an approximate measurement of phase is unfounded.     

The scaling reduction of the three-wave resonant system and the Painlevé VI equation

The scaling invariant solutions of the three-wave resonant system in one spatial and one temporal dimension satisfy a system of three first-order nonlinear ordinary differential equations. These equations can be reduced to one second-order equation quadratic in the second derivative. This equation is outside the class of equations classified by Painlevé and his school. However, it is a special case of an equation recently found to be related via a one-to-one transformation to the Painlevé VI equation.     

The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.     

The discrete variational principle and the first integrals of Birkhoff systems＊

This paper shows that first integrals of discrete equation of motion for Birkhoff systems can be determined explicitly by investigating the invariance properties of the discrete Pfaffian. The result obtained is a discrete analogue of theorem of Noether in the calculus of variations. An example is given to illustrate the application of the results.     

The solution of the cosmological constant problem from the inhomogeneous equation of state — a hint from modified gravity?

The cosmological constant problem is studied in a two component cosmological model. The universe contains a cosmological constant of an arbitrary size and sign and an additional component with an inhomogeneous equation of state. It is shown that, in a proper parameter regime, the expansion of the universe with a large absolute value of the cosmological constant may asymptotically tend to de Sitter space corresponding to a small effective positive cosmological constant. It is argued that such a behavior can be regarded as a solution of the cosmological constant problem in this model. The mechanism behind the relaxation of the cosmological constant is discussed. A connection with modified gravity theories is discussed and an example of a possible realization of the cosmological constant relaxation in f(R) modified gravity is described.     

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we revisit the connection between the Riemann–Roch theorem and the zero-energy solutions of the two-dimensional Dirac equation in the presence of a delta-function-like magnetic field. Our main result is the resolution of a paradox—the fact that the Riemann–Roch theorem correctly predicts the number of zero-energy solutions of the Dirac equation despite counting what seem to be functions of the wrong type.     

First-principles study of the structural,mechanical and electronic properties of ZnX204 （X=Al,Cr and Ga）

This paper performs first-principles calculations to study the structural, mechanical and electronic properties of the spinels ZnA1204, ZnGa2O4 and ZnCr2O4, using density functional theory with the plane-wave pseudopotential method. Our calculations are in good agreement with previous theoretical calculations and the available experimental data. The studies in this paper focus on the evolution of the mechanical properties of ZnAl2O4, ZnGa2O4 and ZnCr2O4 under hydrostatic pressure. The results show that the cubic phases of ZnAl2O4, ZnCa2O4 and ZnCr2O4 become unstable at about 50 GPa, 40 GPa and 25 GPa, respectively. From analysis of the band structure of the three compounds at equilibrium volume, it obtains a direct band gap of 4.35 eV for ZnA1204 and 0.89 cV for ZnCr2O4, while ZnGa2O4 has an indirect band gap of 2.73 eV.     

The Hamiltonian dynamics of the soliton of the discrete nonlinear Schrödinger equation

Hamiltonian equations are formulated in terms of collective variables describing the dynamics of the soliton of an integrable nonlinear Schrödinger equation on a 1D lattice. Earlier, similar equations of motion were suggested for the soliton of the nonlinear Schrödinger equation in partial derivatives. The operator of soliton momentum in a discrete chain is defined; this operator is unambiguously related to the velocity of the center of gravity of the soliton. The resulting Hamiltonian equations are similar to those for the continuous nonlinear Schrödinger equation, but the role of the field momentum is played by the summed quasi-momentum of virtual elementary system excitations related to the soliton.     

The Painlevé property,a modified Boussinesq equation and a modified Kadomtsev-Petviashvili equation

It is shown that the modified Boussinesq equation and the modified Kadomtsev-Petviashvili equation, both of which are known to be completely integrable, possess the Painlev'e property for differential equations as defined by Weiss, Tabor and Carnevale.     

The C2π and the X2σ States of BaF

An extensive calculation was carried out using the precisely measured value of the Q-band heads and the following molecular constants for the C²π¹ ₂ and the X²σ states, which are responsible for the Green Band System of BaF, were determined.