Asymptotic Properties of Solvable $\mathcal{PT}$-Symmetric Potentials

The asymptotic region of potentials has strong impact on their general properties. This problem is especially interesting for PT\mathcal{PT}-symmetric potentials, the real and imaginary components of which allow for a wider variety of asymptotic properties than in the case of purely real potentials. We consider exactly solvable potentials defined on an infinite domain and investigate their scattering and bound states with special attention to the boundary conditions determined by the asymptotic regions. The examples include potentials with asymptotically vanishing and non-vanishing real and imaginary potential components (Scarf II, Rosen-Morse II, Coulomb). We also compare the results with the asymptotic properties of some exactly non-solvable PT\mathcal{PT}-symmetric potentials. These studies might be relevant to the experimental realization of PT\mathcal{PT}-symmetric systems.     

Deuteron radial moments for renormalized chiral potentials

We calculate deuteron positive and negative radial moments involving any bilinear function of the deuteron S and D wave functions for renormalized OPE and TPE chiral potentials. The role played by the strong singularities of the potentials at the origin and the short-distance insensitivity of the results when the potentials are fully iterated is emphasized as compared to realistic potentials.     

The case for the cosmological constant

I present a short overview of current observational results and theoretical models for a cosmological constant. The main motivation for invoking a small cosmological constant (or A-term) at the present epoch has to do with observations of high redshift Type Ia supernovae which suggest an accelerating universe. A flat accelerating universe is strongly favoured by combining supernovae observations with observations of CMB anisotropies on degree scales which give the ‘best-fit’ values Θ_A ⋍ 0.7 and Θ _m ⋍ 0.3. A time dependent cosmological A-term can be generated by scalar field models with exponential and power law potentials. Some of these models can alleviate the ‘fine tuning’ problem which faces the cosmological constant.     

Viscous fluid cosmological models in LRS Bianchi type V universe with varyingΛ

Some LRS Bianchi type V viscous-fluid cosmological models are investigated, in which the coefficient of shear viscosity is considered as proportional to the scale of expansion in the model. This leads toA=Bn, whereA andB are metric potentials,n being a constant. The coefficient of bulk viscosity is also assumed to be a power function of mass density. The cosmological constant is found to be a decreasing function of time, which is supported by results from recent type Ia supernovae observations. Some physical aspects of the models are also discussed.     

LRS Bianchi Type II Bulk Viscous Fluid Universe with?Decaying Vacuum Energy Density Λ

The present study deals with spatially homogeneous and locally rotationally symmetric (LRS) Bianchi type II cosmological models with bulk viscous fluid distribution of matter and decaying vacuum energy density Λ. To get the deterministic models of the universe, we assume that the expansion (θ) in the model is proportional to the shear (σ). This leads to condition R=mS ⁿ , where R and S are metric potentials, m and n are constants. We have obtained two types of models of the universe for two different values of n. The vacuum energy density Λ for both models is found to be a decreasing function of time and it approaches a small positive value at late time which is supported by recent results from the observations of (SN Ia). Some physical and geometric behaviour of these models are also discussed.     

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N ⁻¹. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.     

Interaction potentials,spectroscopy and transport properties of RG+–He (RG=Ar–Rn)

High-level ab initio potential energy curves are calculated for the RG⁺–He complexes (RG=Ar–Rn). RCCSD(T) calculations are employed with large basis sets, and taking account of spin–orbit coupling. The calculated spectroscopic parameters are compared with experimentally determined values, with other high-level ab initio results, and with results from potentials that were obtained by fitting to experimental data. The gas-phase mobilities of RG⁺ ions in He are calculated from our potentials and compared, graphically and statistically, with the experimental mobilities as a function of E/n ₀ at several temperatures. We conclude that more precise experimental data are required in order to discriminate between potentials with more certainty. In addition, we discuss previously reported, unexpectedly large drops in experimental mobility values for RG⁺ in He at 4.35 K as E/n ₀ → 0.     

First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator

We consider 1-D Schr?dinger operators on L ²(R ₊) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved. Received: 16 June 2000 / Accepted: 11 August 2000     

Structure of low-lying <Superscript>12</Superscript>C resonances

The hyperspherical adiabatic expansion is combined with complex scaling and used to calculate low-lying nuclear resonances of ¹²C in the 3α model. We use Ali-Bodmer potentials and compare results for other potentials α-α with similar ⁸Be properties. A three-body potential is used to adjust the ¹²C resonance positions to desired values extending the applicability of the method to many-body systems decaying into three α-particles. For natural choices of three-body potentials we find 14 resonances below the proton separation threshold, i.e. two 0⁺, three 2⁺, two 4⁺, one of each of 1^±, 2^-, 3^±, 4^-, and 6⁺. The partial-wave decomposition of each resonance is calculated as a function of the hyperradius. Strong variation is found from small to large distance. The connection to previous experimental and theoretical results is discussed and agreements as well as disagreements are emphasized.     

Effect of two body core on the charge form factor of3He including three body force and meson exchange current

We compare the effect of two body core on the charge form factor of³He by the hyperspherical harmonics expansion method using various 2BF potentials with the inclusion of three body force. We also include the meson exchange current contribution to the CFF for the same potentials in addition to the 3BF. The results indicate that the combined effect of 3BF and MEC (i) movesq _min ² (the first diffraction minimum) appreciably to the left, amount of shift depends on the 2BF atr ₁₂∼0.7 fm and (ii) enhancesF _max (the height of the secondary maximum of CFF) by an appreciable amount, the increment in general increases with the repulsive core of 2BF (r⩽0.1 fm).     

Interaction potentials,spectroscopy and transport properties of C+(2PJ) and C+(4PJ) with helium

We calculate accurate interatomic potentials for the interaction of a singly charged carbon cation with a helium atom. We employ the RCCSD(T) method, and basis sets of quadruple-ζ and quintuple-ζ quality; each point is counterpoise corrected and extrapolated to the basis set limit. We consider the two lowest C⁺(²P) and C⁺(⁴P) electronic states of the carbon cation, and calculate the interatomic potentials for the terms that arise from these: ²Π and ²Σ⁺, and ⁴Π and ⁴Σ^?, respectively. We additionally calculate the interatomic potentials for the respective spin–orbit levels, and examine the effect on the spectroscopic parameters. Finally, we employ each set of potentials to calculate transport coefficients, and compare these to the available data. Critical comments are made in the cases where there are discrepancies between the calculated values and measured data.     

A solution for galactic disks with Yukawian gravitational potential

We present a new solution for the rotation curves of galactic disks with gravitational potential of the Yukawa type. We follow the technique employed by Toomre in 1963 in the study of galactic disks in the Newtonian theory. This new solution allows an easy comparison between the Newtonian solution and the Yukawian one. Therefore, constraints on the parameters of theories of gravitation can be imposed, which in the weak field limit reduce to Yukawian potentials. We then apply our formulae to the study of rotation curves for a zero-thickness exponential disk and compare it with the Newtonian case studied by Freeman in 1970. As an application of the mathematical tool developed here, we show that in any theory of gravity with a massive graviton (this means a gravitational potential of the Yukawa type), a strong limit can be imposed on the mass (m _g) of this particle. For example, in order to obtain a galactic disk with a scale length of b∼ 10 kpc, we should have a massive graviton of m _g ≪ 10⁻⁵⁹g. This result is much more restrictive than those inferred from solar system observations.     

On the Born-Oppenheimer expansion for polyatomic molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh ² of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.     

Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Pöschl–Teller potentials in terms of their degree ℓ polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (ℓ=1,2,…) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gómez-Ullate et al.'s are the first members of these infinitely many potentials.     

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L ^p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L ² case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some L^s_weak{L^{s}_{weak}} or L ^s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.     

LRS Bianchi Type II Perfect Fluid Cosmological Models in Normal Gauge for Lyra’s Manifold

The present study deals with spatially homogeneous and locally rotationally symmetric (LRS) Bianchi type II cosmological models of perfect fluid distribution of matter for the field equations in normal gauge for Lyra’s manifold where gauge function β is taken as time dependent. To get the deterministic models of the universe, we assume that the expansion (θ) in the model is proportional to the shear (σ). This leads to condition R=mS ⁿ , where R and S are metric potentials, m and n are constants. We have obtained two types of models of the universe for two different values of n. It has been found that the displacement vector β behaves like cosmological term Λ in the normal gauge treatment and the solutions are consistent with recent observations. Some physical and geometric behavior of these models are also discussed.     

自由能方法与零压下Al的熔化温度

利用自由能方法的分子动力学模拟,计算了零压下Al的熔化温度.在计算液相自由能的过程中,采用勒纳-琼斯(LJ)液体作为参考系统,同时将计算结果与Mei和Davenport等人的计算结果进行了比较,计算结果表明：1)选用LJ参考系统使液相自由能的计算时间节省一半,并且不影响熔化温度的计算结果;2)采用不同的埋入原子势(EAM)的分子动力学模拟计算得到的熔化温度与实验值都存在偏差,而就金属Al而言,采用Cai等人的EAM势的熔化温度的计算结果比Mei和Davenport及Morris等人采用的势模型的结果略有改 关键词： 熔化温度 自由能方法 分子动力学模拟     

Theoretical study of Si+(2PJ)–RG complexes and transport of Si+(2PJ) in RG (RG = He–Ar)

We calculate accurate interatomic potentials for the interaction of a singly charged silicon cation with a rare gas atom of helium, neon or argon. We employ the RCCSD(T) method, and basis sets of quadruple-ζ and quintuple-ζ quality; each point is counterpoise-corrected and extrapolated to the basis set limit. We consider the lowest electronic state of the silicon atomic cation, Si⁺(²P), and calculate the interatomic potentials for the terms that arise from this: ²Π and ²Σ⁺. We additionally calculate the interatomic potentials for the respective spin-orbit levels, and examine the effect on the spectroscopic parameters; we also derive effective ionic radii for C⁺ and Si⁺. Finally, we employ each set of potentials to calculate transport coefficients, and compare these to available data for Si⁺ in He.     

Pumping current of a Luttinger liquid with finite length

We study transport properties in a Tomonaga-Luttinger liquid in the presence of two time-dependent point like weak impurities, taking into account finite-length effects. By employing analytical methods and performing a perturbation theory, we compute the backscattering pumping current (I _bs ) in different regimes which can be established in relation to the oscillatory frequency of the impurities and to the frequency related to the length and the renormalized velocity (by the electron-electron interactions) of the charge density modes. We investigate the role played by the spatial position of the impurity potentials. We also show how the previous infinite length results for I _bs are modified by the finite size of the system.     

Bound state solutions of d-dimensional Schrödinger equation with Eckart potential plus modified deformed Hylleraas potential

We study the d-dimensional Schrödinger equation for Eckart plus modified deformed Hylleraas potentials using the generalized parametric form of Nikiforov-Uvarov method. We obtain energy eigenvalues and the corresponding wave function expressed in terms of Jacobi polynomial. We also discuss two special cases of this potential comprised of the Hulthen potential and the Rosen-Morse potential in three dimensions. Numerical results are also computed for the energy spectrum and the potentials.