Integral equation theory of the exchange potential,HOMO–LUMO properties,and sum rules for the exchange-correlation force

Recently, a formally exact integral equation for the exchange potential V _x(r) has been presented by the authors. In the admittedly simplistic limit in which Slater–Kohn–Sham and Hartree–Fock determinants become equal, this integral equation reduces to that given by Della Sala and Görling. Here, a proposal is made to relate, but now approximately, the formally exact equation for V _x(r) to HOMO–LUMO properties. The addition of a correlation contribution V _c(r) to V _x(r), the sum being the exchange-correlation potential V _xc(r), is finally considered, some exact properties and especially sum rules for the force ??V _xc/?r being the focus.     

The generalized exponential-integral V(x,y)=∫1∞ exp(−xt)ln(t+y)dtt and computer algorithms for y = 0, 1

The generalized exponential-integral function V(x, y) defined here includes as special cases the function E⁽²⁾₁(x) = V(x, 0) introduced by van de Hulst and functions M₀(x) = V(x, 1) and N₀(x) = V(x, -1) introduced by Kourganoff in connection with integrals of the form ∫ E_n)t)E_m(t±x), which play an important role in the theory of monochromatic radiative transfer. Series and asymptotic expressions are derived and, for the most important special cases, y = 0 and y = 1, Chebyshev expansions and rational approximations are obtained that permit the function to be evaluated to at least 10 sf on 0<x<∞ using 16 sf arithmetic.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

The 1/r expansion for the critical multiple well problem

We consider the critical multiple well problem $$H = - \Delta + \sum\limits_{i = 1}^n {V(x - rx_i )} ,$$ where ?Δ+V(x) has a zero energy resonance. We prove that all eigenvalues and resonances ofH tending to zero as 1/r ² are analytic in 1/r. We give an explicit equation for the lowest nonvanishing coefficient in the 1/r expansion for any of these eigenvalues or resonances and observe thatH has infinitely many resonances tending to zero. Forn=2 andn=3, we compute the coefficients explicitly and forn=2, we also give the next coefficient in the 1/r expansion.     

On the Fundamental Solution of a Linearized Homogeneous Coagulation Equation

We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K(x, y) = (xy)^λ/2, around the steady state f(x) = x ^?(3+λ)/2 with ${\lambda \in (1, 2)}We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K(x, y) = (xy)^λ/2, around the steady state f(x) = x ^−(3+λ)/2 with l ? (1, 2){\lambda \in (1, 2)} . Detailed estimates on its asymptotics are obtained. Some consequences are deduced for the flux properties of the particles distributions described by such models.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

Bounded perturbations of the P(ϕ)2 theory

As a perturbation to the P(?)₂ theory we consider interaction densities of the form V(?(x)), where ?(x) is a scalar hermitian boson field and V(α) is a bounded real continuous function. It is proved that the asymptotic fields exist and are equal to the asymptotic fields of the P(?)₂ theory. The connection with non-polynomial theories of rational type is indicated. Furthermore the consequences of a bounded perturbation for the S-matrix and the spectral properties are discussed.     

The Ruderman-Kittel-Kasuya-Yosida interaction in amorphous La80Au20 alloys with dilute Gd impurities

From magnetization measurements on some amorphous dilute La_80?xGd_xAu₂₀ alloys with x ? 1 we have shown that the magnetic behavior follows the scaling laws of a spin-glass system, characteristic of the 1/r³ dependence of the pairwise interaction. We have also determined the strength of the Ruderman-Kittel-Kasuya-Yosida interaction V(r) = (V₀cos 2k_Fr)/r³, to be V₀ = 0.20 × 10^?37 ergcm³. The corresponding value of the s-f exchange integral is |J_sf| = 0.14 eV, which is compared with values determined from other experiments.     

Predictions for inclusive reactions based on a parton structure of hadrons

It is proposed that the asymptotic single-particle distributions f(x)=x(dσ/dx) in inclusive reactions should reflect the limiting longitudinal momentum distribution, ∼νW₂(x), of the hypothetical partons constituting a hadron. It is further argued that f(x) becomes, proportional to νW₂(x) as x → 1. Predictions are then made for a large class of inclusive reactions based on our knowledge of νW₂(x) from electroproduction data. For some of these (such as pp → π^±X), in which the predictions differ markedly from the Regge ones, present experiments distinctly favor ours.     

Quark charges and colour gluon mass from deep-inelastic bremsstrahlung

We derive sum rules for the structure functionV(x) for the ‘three-photon’ processe ^±+p→e ^±+γ+X which can distinguish between various colour models below colour threshold, independently of the quark and gluon distributions. A careful study of the sum rule forV(x) in the broken colour gauge theory model can in principle be used to determine the colour gluon mass. Invoking the specific assumptions of the dominance ofp-type quarks and neglecting the sea of quark-antiquark pairs, we also obtain bounds forV(x) in terms ofvW ₂ (x) which can distinguish between various colour models below colour threshold.     

Description of ultracold atoms in a one-dimensional geometry of a harmonic trap with a realistic interaction

We compute the ground-state energy of two atoms in a one-dimensional geometry of a harmonic optical trap. We obtain a dependence of the energy on a one-dimensional scattering length, which corre-sponds to various strengths of the interaction potential V _int (x) = V ₀ exp {?2cx ²}. The calculation is performed by numerical and analytical methods. For the analytical method we choose the oscillator representation method (OR), which has been successfully applied to computations of bound states of various few-body systems. The main results of this paper are (1) a numerical investigation of the validity range of the previously used pseudopotential method and (2) an investigation of the validity range of the OR for the potential V(x) = V _conf (x) + V _int (x) = x ²/2 + V ₀ exp {?2cx ²}.     

An approach to radially symmetrical solutions of the sine-Gordon equation

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2π-kink, in the expanding and in the shrinking phase, for R(t)j? R(0). It is shown that the parameterization φ(r, t) = 4 arcian exp[γ(r?R(0)] + x(r, t), where R(t) describes the exact propagation of the maximum of φ,(r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x(r = R(t), t) is found and tested numerically. A relationship between the fluctuations in x(r = R(t), t) and those in $\text{R}\text{?}\text{(t), t)}\text{and}\text{R(t)}$ explains why the solitary wave is almost stable. From x(r = R(t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ_r(r, t) with respect to r = R(t) is predicted. It also exhibits fluctuations corresponding to those in x(r = R(t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.     

Effects of nickel doping on structural and optical properties of spinel lithium manganate thin films

Spinel LiNi_xMn_2−xO₄ (x≤0.9) thin films were synthesized by a sol-gel method employing spin-coating. The Ni-doped films were found to maintain cubic structure at low x but to exhibit a phase transition to tetragonal structure for x≥0.6. Such cubic-tetragonal phase transition can be explained in terms of Ni³⁺(d⁷) ions with low-spin (t_2g⁶,e_g¹) configuration occupying the octahedral sites of the compound, thus being subject to the Jahn-Teller effect. By X-ray photoelectron spectroscopy both Ni³⁺ and Ni²⁺ ions were detected where Ni²⁺ is more populated than Ni³⁺. Optical properties of the LiNi_xMn_2−xO₄ films were investigated by spectroscopic ellipsometry in the visible-ultraviolet range. The measured dielectric function spectra mainly consist of broad absorption structures attributed to charge-transfer transitions, O²⁻(2p)→Mn⁴⁺(3d) for 1.9 (t_2g) and 2.8-3.0 eV (e_g) structures and O²⁻(2p)→Mn³⁺(3d) for 2.3 (t_2g) and 3.4-3.6 eV (e_g) structures. Also, sharp absorption structures were observed at about 1.6, 1.7, and 1.9 eV, interpreted as being due to d-d crystal-field transitions within the octahedral Mn³⁺ ion. In terms of these transitions, the evolution of the optical absorption spectrum of LiMn₂O₄ by Ni doping could be explained and the related electronic structure parameters were obtained.     

A new treatment of singular potentials

For a broad class of strongly singular potentials, the effective (finite-dimensional) forms of the hamiltonian H are constructed by its “smooth” algebraic truncation in a large model space. The method is based on an asymptotic factorization of H into a product of matrices. Its efficiency (acceleration of convergence of the energies) is illustrated on the singularly anharmonic potential V(r) = r² + hr^?4.     

Ionization of Coulomb Systems in {\mathbb{R}^3} by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iy_t=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi}, x ? \mathbbR³{x\in\mathbb{R}^3}, r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.     

Electric field induced deformation of hemispherical sessile droplets of ionic liquid

Sessile droplets of an ionic liquid with contact angles close to 90° were subjected to an electric field E = V/w inside a capacitor with plate separation w and potential difference V. For small field induced deformations of the droplet shape the change in maximum droplet height, Δh = h(E) – h(0), was found to be virtually independent of the plate separation provided that w > 3h(0). In this regime a scaling law obtains Δh α E²r², where r is the constant droplet radius, in agreement with the asymptotic predictions of Basaran and Scriven (J. Coll. Int. Sci. 140, 10, 1990).