A solution spectrum of the nonlinear Schrödinger equation. II

It was shown in a previous communication that the nonlinear Schrödinger equation exhibits a spectrum of eigenfunctions of the form = _k,A _k (coshkx) ^–k and = _k B _k (coshkx) ^–k–1sinhkx, and the corresponding eigenvalues of the energy are related to a band structure with a characteristic energy gap as a significant feature. In the present paper, it is shown that a further spectrum exists exhibiting the general structure = _k=0 A _k(cosh kx)^–k–1/2and = _k=0 B_k(cosh kx)^–k–3/2sinhkx and yielding also a band structure. An extension of the solution spectrum to a nonlinear Klein-Gordon equation and a nonlinear Dirac equation does not imply essential difficulties, and the corresponding characteristic band structure has to be related to a mass spectrum.     

Bcklund transformations for the Burgers equation via localization of residual symmetries

We obtain the non-local residual symmetry related to truncated Painlev′e expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system.By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly,we also localize the linear superposition of multiple residual symmetries to find the corresponding finite transformations.It is interesting to find that the n-th B¨acklund transformation for Burgers equation can be expressed by determinants in a compact way.     

On the Uniqueness of Gibbs Measures for <Emphasis Type="Italic">p</Emphasis>-Adic Nonhomogeneous λ-Model on the Cayley Tree

We consider a nearest-neighbor p-adic -model with spin values ±1 on a Cayley tree of order k 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p 3. If p=2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.     

Integral form of the time-dependent radiation transfer equation—II. Inhomogeneous spherical media

The integral form of the radiation transfer equation, with spherical symmetry for a non-isotropic, time-dependent, inhomogeneous, non-scattering medium is given for arbitrary initial and boundary conditions. The optical depth is generalized to an optical retardation and a separation parameter is introduced to separate boundary and initial conditions. The static solution is identical to the asymptotic form of the time-dependent solution.     

Evolution of solitons described by the higher-order nonlinear Schrödinger equation. II. Numerical investigation

The resonant radiation and evolution of a soliton described by the higher-order nonlinear Schrödinger equation is estigated numerically. The results are in good agreement with the analytical theory [V.I. Karpman, Phys. Lett. A 244 698) 397].     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).     

The infra-red L.M.R. spectrum of SeH in the X 2Π state

Magnetic dipole transitions between the ²Π_1/2 and ²Π_3/2 components of the ground state of SeH have been observed at about 1780 cm^-1 in a L.M.R. experiment using a CO laser. Transitions have been detected for the four most abundant natural isotopes of Se. The transition frequencies depend primarily on the magnitude of the spin-orbit coupling constant and have been analysed in conjunction with data from the E.P.R. spectrum to yield

The admixture of other electronic states by spin-orbit coupling has been investigated to third order in perturbation theory and found to produce significant effects.     

On the Lippmann-Schwinger equation in Liouville space. Π&#x030A; subdynamics

We study some aspects of the Lippmann-Schwinger equation in Liouville space: The relation between the Møller approach and Π&#x030A; subdynamics, the obtention of H-invariants from the corresponding H₀-invariants and the derivation of kinetic equations.     

To the explanation of the “white” line in the X-ray <Emphasis Type="Italic">K</Emphasis>-absorption spectrum of sulfur in NaBiS<Subscript>2</Subscript>

The maximum in the K-absorption spectra of sulfur in LiBiS₂, NaBiS₂, and KBiS₂, which is located more than 12 eV above the absorption edge and is especially strong for NaBiS₂, cannot be explained within single-electron calculations. Apparently, it is not related to any ordering in the arrangement of alkali metal and bismuth atoms in the cation sublattice. A scheme is considered in which a Na₂S-Bi₂S₃ solid solution is formed with subsequent oxidation of Na₂S with the formation of sulfate Na₂SO₄ with sulfur in the valence state +6, as a result of which the absorption edge shifts by approximately 13 eV. Thus, the initial experimental spectrum is treated as a weighted sum of the K-absorption spectra of sulfur in NaBiS₂, Na₂S, Bi₂S₃, and Na₂SO₄.     

Arbitrary <Emphasis Type="Italic">l</Emphasis>-state solutions of the Schrödinger equation for the Eckart potential with an improved approximation of the centrifugal term

Applying an improved approximation scheme to the centrifugal term, the approximate analytical solutions of the Schrödinger equation for the Eckart potential are presented. Bound state energy eigenvalues and the corresponding eigenfunctions are obtained in closed forms for the arbitrary radial and angular momentum quantum numbers, and different values of the screening parameter. The results are compared with those obtained by the other approximate and numerical methods. It is shown that the present method is systematic, more efficient and accurate.     

The Parameter Planes of λz<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> exp(z) for <Emphasis Type="Italic">m</Emphasis> ≥ 2*

We consider the families of entire transcendental maps given by F _λ,m (z) = λz ^m exp(z), where m ≥ 2. All functions F _λ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A ^*(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A ^*(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of which serve as a model for F _λ,m , in the sense that they are related by means of quasiconformal surgery to F _λ,m . Both authors were supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028. The first author was also supported by MTM2006-05849/Consolider (including a FEDER contribution).     

Reconstruction of the 2D hole gas spectrum for selectively doped <Emphasis Type="Italic">p</Emphasis>-Ge/Ge<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>S<Subscript>ix</Subscript> heterostructures

The magnetic field (0≤B≤32 T) and temperature (0.1≤T≤15 K) dependences of longitudinal and Hall resistivities have been investigated for p-Ge_0.93Si_0.07/Ge multilayers with different Ge layer widths 12≤d _w ≤20 nm and hole densities p _s =(1–5)×10¹⁵ m^?2. An extremely high sensitivity of the experimental data (the structure of magnetoresistance traces, relative values of the inter-Landau-level gaps deduced from the activation magnetotransport, etc.) to the quantum well profile is revealed in the cases where the Fermi level reaches the second confinement subband. An unusually high density of localized states between the Landau levels is deduced from the data. Two models for the long-range random impurity potential (the model with randomly distributed charged centers located outside the conducting layer and the model of the system with a spacer) are used to evaluate the impurity potential fluctuation characteristics: the random potential amplitude, the nonlinear screening length in the vicinity of integer filling factors v=1 and v=2, and the background density of states (DOS). The described models are suitable for explanation of the observed DOS values, while the short-range impurity potential models fail. For half-integer filling factors, a linear temperature dependence of the effective quantum Hall effect plateau-plateau (PP) transition widths v₀(T) is observed, contrary to the expected scaling behavior of the systems with short-range disorder. The finite T→0 width of the PP transitions may be due to an effective low-temperature screening of a smooth random potential due to the Coulomb repulsion of electrons.     

Dirac equation with spin symmetry for the modified Pöschl–Teller potential in <Emphasis Type="Italic">D</Emphasis> dimensions

We present solutions of the Dirac equation with spin symmetry for vector and scalar modified Pöschl–Teller potentials within the framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov–Uvarov method and the two-component spinor wave functions obtained are in terms of the Jacobi polynomials. It is found that there exist only positive energy states for bound states under spin symmetry, and the energy of a level with fixed value of n, increases with increase in dimension of space time and the potential range parameter α.     

Alternative method of calculating the eigenvalues of the transfer matrix of the τ<Subscript>2</Subscript> model for <Emphasis Type="Italic">N</Emphasis> = 2

It has been shown that the τ₂ (Baxter-Bazhanov-Stroganov) model for N = 2 with arbitrary parameters is a particular case of the generalized Ising model. The model satisfies the free-fermion condition, which enables one to solve it by the method of the auxiliary Grassmann field. Explicit expressions have been derived for the partition function on a finite-size lattice and eigenvalues of the transfer matrix. In this approach, in contrast to the functional relation method, there is no problem with the multiplicities of the eigenvalues of transfer matrix.     

Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any <Emphasis Type="Italic">κ</Emphasis>-value

The Dirac equation, with position-dependent mass, is solved approximately for the generalized Hulthén potential with any spin-orbit quantum number κ. Solutions are obtained by using an appropriate coordinate transformation, reducing the effective mass Dirac equation to a Schrödinger-like differential equation. The Nikiforov-Uvarov method is used in the calculations to obtain energy eigenvalues and the corresponding wave functions. Numerical results are compared with those given in the literature. Analytical results are also obtained for the case of constant mass and the results are in good agreement with the literature.     

Arbitrary-state solutions of the Dirac equation for a Möbius square potential using the Nikiforov-Uvarov method

We inquire into spin and pseudospin symmetries of the Dirac equation under a Mbius square-type potential using the Nikiforov-Uvarov method to calculate the bound state solutions. We numerically discuss the problem and include various explanatory figures.     

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type. II. The <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">N</Emphasis>−1</Subscript> Case: First Steps

This is the second part of a series of papers concerning Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. We present an explicit diagonalization of special Hilbert-Schmidt operators arising in the free A _N−1 case, and a spectral structure analysis of the commuting family of Hilbert-Schmidt operators associated with the general A _N−1 case.     

Forbidden 2<Emphasis Type="Italic">P</Emphasis>–<Emphasis Type="Italic">nP</Emphasis> and 2<Emphasis Type="Italic">P</Emphasis>–<Emphasis Type="Italic">nF</Emphasis> transitions in the energy spectrum of ultracold Rydberg lithium-7 atoms

Forbidden 2P–nP and 2P–nF transitions in the ranges of the principal quantum number n = 42–114 and n = 38–48 have been detected in the optical spectra of ultracold highly excited lithium-7 atoms. The presence of forbidden transitions is due to induced external electric fields. The quantum defects and ionization energy obtained in various experiments and predicted theoretically have been discussed.