A novel strong coupling expansion of the QCD Hamiltonian

Introducing an infinite spatial lattice with box length a, a systematic expansion of the physical QCD Hamiltonian in λ = g ^?2/3 can be obtained, with the free part being the sum of the Hamiltonians of the quantum mechanics of spatially constant fields for each box, and interaction terms proportional to λ ⁿ with n spatial derivatives connecting different boxes. As an example, the energy of the vacuum and the lowest scalar glueball is calculated up to order λ ² for the case of SU(2) Yang-Mills theory.     

An analysis of the “tail” of the series in density for the Coulomb crystal energy

The crystal energy of the metallic hydrogen is found within the framework of the Wigner-Seitz method, in the form of the power series in the density parameter r_s. The series appears to be absolutely convergent within the radius r₀ = 4.2a_B. The correspondence between this series and the perturbation theory expansion is established for terms of 2nd, 3rd, and 4th order. The value of the “tail” is estimated from the sum of all higher order terms of the first series.     

Wave-spreading effects in nuclear scattering at intermediate energies

Based on the Fresnel approach discussed in a previous article the S-matrix for nuclear elastic scattering is factorized into an infinite product, the n-th term of which is essentially given by (n ? 1)-fold commutators of the interaction potential at n different z-coordinate positions. A subsequent cumulant expansion expresses the Fresnel phase shift as an infinite sum of many-body clusters. In constrast to eikonal-type expansions each term of this sum contains contributions of all orders in the inverse wave number. Because of this feature the present expansion is useful for scattering from light nuclei over a wide range of energies and scattering angles.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger representation space that incorporates an analytic solution for the non-central electric dipole potential cosθ/r², which was believed not to belong to the class of exactly solvable potentials. Therefore, we were able to obtain a closed form solution of the three-dimensional time-independent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series in a basis composed of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.     

Sum rules for the bound-free transition form factors in hydrogenlike atoms

The product of two bound-free transition form factors of the type 〈k¦exp(i q·r)¦nlm〉 is analytically summed-up over all the degenerate stateslm. Using the genuine Coulomb wave 〈r¦k〉, together with the hydrogenlike wavefunction 〈r¦nlm〉, exact analytical expressions are derived for the sum rule. The results are conveniently expressed in terms of the finite linear combination of easily generated Appell and Lauricella hypergeometric polynomials of two and three variables. This highly desirable sum rule is most frequently required in broad applications within the distorted wave theory of particle-atom scattering.     

Self-consistent Overhauser model for the pair distribution function of an electron gas at finite temperature

We present calculations of the spin-averaged pair distribution function g(r) in a homogeneous gas of electrons moving in dimensionality D=3 or D=2 at finite temperature. The model involves the solution of a two-electron scattering problem via an effective potential, which embodies many-body effects through a self-consistent Hartree approximation, leading to two-body wave functions to be averaged over a temperature-dependent distribution of relative momentum for electron pairs. We report illustrative numerical results for g(r) in an intermediate-coupling regime and interpret them in terms of changes of short-range order with increasing temperature.     

A new boundary method for electromagnetic scattering from inhomogeneous bodies

A new numerical method for scattering from inhomogeneous bodies is presented. In particular, the 2D case of a TM-polarizated incident wave scattered by an infinite cylinder is considered. The scattered field is sought in two different domains. The first one is a bounded region inside the scattering body with an inhomogeneous permittivity ε(x,y). The second one is an unbounded homogeneous region outside the scatterer. An approximate solution for the scattered field inside the scatterer is sought by applying the QTSM technique. The method of discrete sources is used to approximate the scattered field in the unbounded region outside the scattering body. A comparison of the numerical solution with an analytic solution is performed.     

Boson expansions and quantization of time-dependent self-consistent fields (I). Particle-hole excitations

Two concrete methods are presented for quantizing the time-dependent Hartree equations in terms of boson operators. The first is the well-known infinite boson expansion analogous to the Holstein-Primakoff representation of angular momentum operators. The second, a new development, consists of finite boson quadratic forms, and is analogous to the Schwinger representation of angular momenta. In each case, a physical boson subspace can easily be constructed within which the full fermion dynamics is exactly duplicated. It therefore follows that quantization of the time-dependent Hartree equations, including all degrees of freedom, retrieves the exact many-body problem. The discussion in this paper is limited to particle-hole excitations of an N-particle system. A generalization to one-nucleon transfer processes on the N-particle system is also given in terms of ideal odd nucleons, but this brings in infinite expansions.     

Solution of Schrödinger and related equations for irregular and composite regions

The Schrödinger equation with a potential is mathematically equivalent to the Helmholtz equation with a spatially variable propagation constant. A new method is presented for solving certain standard problems associated with these equations. In the Schrödinger language these are the ones in which a potential ν(|r|) acts inside an irregular (aspherical) boundary, where r has its origin inside the boundary. In terms of the Helmholtz equation, these include problems in which a region of constant index of refraction, but arbitrary shape, is embedded in a second uniform region with a different index. It is shown how bound state and scattering problems for such a potential (or region) can be treated in a way that avoids the usually intractable problem of matching solutions across the irregular boundary. The method requires, in general, the truncation of an infinite set of equations for partial wave amplitudes. The special case is discussed of a potential that becomes infinite throughout a region, so the wave amplitude must vanish inside the region (and, hence, on its boundary). For a long wave length this becomes a problem with the Laplace equation, and the general technique is illustrated by a calculation of the free charge on a perfectly conducting spheroid. The theory is extended from a single potential to an ensemble of such potentials, and in particular to an ensemble of potentials with spherical boundaries. In the special case that the potentials are arranged in a periodic lattice the formulas resemble those obtained by the KKR method, but are simpler in some ways. The method is extended to an ensemble of irregular potentials, and these results are shown to be applicable to the special case of an ensemble of finite range, but overlapping, spherical potentials.     

Atom-phonon interaction at a surface

The interaction between an atom and a phonon is evaluated at a surface. The basic assumption is that the total potential energy of the atom V(r) can be written as the sum of pair interactions w(r?r_i with individual substrate atoms. The result is an expansion in phonon wavevector q, with coefficients expressed in terms of the equilibrium adsorption potential. The coupling can be utilized in calculations of kinetic and equilibrium properties of adsorbed atoms.     

An infinite lie group of symmetry of one-dimensional gas flow,for a class of entropy distributions

We have shown in earlier works the existence of three previously unknown symmetries of the equations of one-dimensional gas dynamics, with arbitrary entropy distribution and arbitrary polytropic index γ. These symmetries are seen here to form a group whenever the equation of state is of the form P = ?³(a₀ + a₁M + a₂M²)^?2 where M = ∝?dr is the Lagrangian mass coordinate.Introducing the remaining symmetry of space-translation enlarges the group into a Lie group of symmetry of infinite order, from which an infinite number of conservation laws can be deduced by application of Noether's theorem. The Lie group has a finite sub-algebra of order eight, which has SU3 structure; the list of associated conservation laws includes each of the six ones that are derivable from general physical principles, namely: the energy, momentum and the center-of-mass integrals, two integrals expressing scale invariance, and one associated with the virial theorem; the remaining two integrals of the octet are of a new type.Such a situation reminds us of the case of the Korteweg-de Vries equation in the soliton problem, where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse-scattering method. Thus the question is raised of whether the inverse-scattering method also applies to gas-dynamical equations (with the above equation of state), or else whether another method of linearization may be found.     

Exact L series solution of the Dirac-Coulomb problem for all energies

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

Approximate eigensolutions of Dirac equation for the superposition Hellmann potential under spin and pseudospin symmetries

The Hellmann potential is simply a superposition of an attractive Coulomb potential ?a/r plus a Yukawa potential be^{?δ r} /r. The generalized parametric Nikiforov–Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number κ in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.     

On standing wave solutions to the Schrödinger equation

The standing wave solution to the Schrödinger equation defined in terms of the standing wave Green's function for the full Hamiltonian is discussed. This solution is compared with the more usual standing wave solution. The former is shown to be one-half the sum of usual ingoing and outgoing wave solutions obeying Lippmann-Schwinger equations. Partial wave elements of the two solutions as well as of the two reaction (K) matrices are found to be related by a simple normalization factor, viz. cos²δ_l, where δ_l is the lth partial wave phase shift. Thus, either of the two standing wave solutions can be used to obtain the correct K matrix element, tan δ_l, since in each case it is the asymptotic ratio of the irregular to the regular solution.