Jaime Zaratiegui
Using a complete basis set we have obtained an analytic expression for the matrix elements of the Coulomb interaction. These matrix elements are written in a closed form. We have used the basis set of the three-dimensional isotropic quantum armonic oscillator in order to develop our calculations, which can be useful when treating interactions in localized systems.
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On the geometry of Coulomb branches
本文对三维\(N = 4\)规范理论的库仑分支进行几何描述，通过环面紧致化爆破统一处理不同情形。利用局部描述分类辛叶并描述横向切片，还讨论了相关主题，如改进定理假设及阐明与横向希尔伯特概型关系。
Geometry-induced Coulomb-like potential in helically twisted quantum systems
本文研究三维螺旋扭曲空间中带非平庸扭转参数的薛定谔方程，通过变量分离得径向方程，发现扭转诱导的动量耦合产生纯几何起源的类库仑势，获束缚态精确解并验证数值一致性，揭示几何形变的量子限制机制。
The Monopole and the Coulomb field as duals within the unifying Reissner-Nordstrom geometry
该研究证明磁单极场与库仑场在Reissner-Nordstrom几何中互为对偶，二者都是构造局域协变对角化应力-能量张量的标架所必需的，同属该几何的固有组成部分。
Dyson has shown an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. In this paper, we introduce finite-range Coulomb gas (FRCG) models as a generalization of the Dyson models with a finite range of eigenvalue interactions. As the range of interaction increases, there is a transition from Poisson statistics to classical random matrix statistics. These models yield new universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems w…
The two-body Coulomb Hamiltonian, when calculated in Coulomb-Sturmian basis, has an infinite symmetric tridiagonal form, also known as Jacobi matrix form. This Jacobi matrix structure involves a continued fraction representation for the inverse of the Green's matrix. The continued fraction can be transformed to a ratio of two $_{2}F_{1}$ hypergeometric functions. From this result we find an exact analytic formula for the matrix elements of the Green's operator of the Coulomb Hamiltonian.
Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems
本文提出一个基于辛几何和相空间约束的几何框架，通过Kustaanheimo-Stiefel正则化与三个几何约束，无需算子对角化即可精确恢复三维库仑束缚态的离散能级和简并度，并应用于杂质态、莫尔超晶格等凝聚态系统。
We analytically evaluate the generating integral $K_{nl}(β,β') = \int_{0}^{\infty}\int_{0}^{\infty} e^{-βr - β' r'}G_{nl} r^{q} r'^{q'} dr dr'$ and integral moments $J_{nl}(β, r) = \int_{0}^{\infty} dr' G_{nl}(r,r') r'^{q} e^{-βr'}$ for the reduced Coulomb Green's function $G_{nl}(r,r')$ for all values of the parameters $q$ and $q'$, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.
Exact analytic expression is derived for the matrix elements of the Coulomb interaction in two dimensions in the form of a closed finite sum expression. The orthonormal complete set of eigenfunctions of the harmonic oscillator is used as the basis for spanning real space. Several recurrence relations have been found in order to simplify the task of calculating the usually vast amount of elements required for any computer simulation.
In a recent paper by Barton (J. Phys. A40, 1011 (2007)), the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V_1(|x|) = -α/(|x|+a). In the present paper, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cutoff Coulomb potential V_2(|x|) = -α/|x|, ~|x| > a, and otherwise V_2(|x|) = -α/a.
We establish sharp deviation inequalities for the linear statistics of the 2D Coulomb gas. These imply sub-Gaussian inequalities, where the variance is given by the Dirichlet norm. The proofs use complex geometry and potential theory on Riemann surfaces and apply more generally to beta-ensembles, which also include integer Quantum Hall states on Riemann surfaces. In a sequel of the paper we give applications to large and moderate deviation principles, local laws at mesoscopic scales, quantitative Bergman kernel asymptotics. In a series of companion papers applications to concentration of measu…
Bound and scattering states of a non-central extension of the three-dimensional Kepler-Coulomb Hamiltonian are worked out analytically within the framework of the potential groups of the problem, SO(7) for bound states and SO(6,1) for scattering states. In the latter case, the S matrix is calculated by the method of intertwining operators.
Sphere quantization of Higgs and Coulomb branches and Analytic Symplectic Duality
本文利用三维超共形场论的受保护球面与半球关联函数量子化 Higgs、Coulomb 分支及其实轨迹，并由局域化与对偶性导出代数幺正表示预测，提出解析形式的辛对偶。
We present the analytical solution in closed form for the semiclassical limit of the quantum mechanical Coulomb Green function in position space in n dimensions. We utilize a projection method which has its roots in Lambert's theorem and which allows us to treat the system as an essentially one dimensional problem. The semiclassical result assumes a simple analytical form and is well suited for a numerical evaluation. The method can also be extended to classically forbidden space regions. Already for moderately large principal quantum numbers nu >= 5, the semiclassical Green function is found …
The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by F~Stillinger has been calculated to be equal to $Z_{B}^{\infty}\ =\ 0.904854$ and $Z_{B}^{m_p}\ =\ 0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $Re Z < Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane.…
We derive the large-N, all order asymptotic expansion for a system of N particles with mean-field interactions on top of a Coulomb repulsion at temperature 1/β, under the assumptions that the interactions are analytic, off-critical, and satisfy a local strict convexity assumption.
We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulten plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudo-centrifugal) term. Using the Nikiforov-Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modifiable of our results into some particular potential cases, u…
A unified formula for analytical evaluation of two-center exchange, hybrid and coulombic type integrals is presented.
Motivated by a recent analysis which presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a ("hard-core amended") Coulomb potential in one dimension. We prove the existence of a critical coupling constant (which contradicts the assertions of the previous analysis) and give analytic upper bounds on the energy eigenvalues. These upper bounds seem to disprove the previous explicit solution.
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