Convergence properties of Fock's expansion for S-state eigenfunctions of the helium atom

It is proved by functional analytic methods that for S-state solutions of Schrödinger's equation for the helium atom, Fock's expansion in powers of R ^1/2 and R ln R, where R is the hyperspherical radius r ₁ ² +r ₂ ² , converges pointwise for all R, thereby generalising a result of Macek that the expansion converges in the mean for all R<1/2. It is shown that for any value (even complex) of the energy E, Schrödinger's equation, considered as a partial differential equation with no boundary condition at R=, has infinitely many solutions representable by an expansion of the type proposed by Fock. Some of the open problems are discussed in determining whether for E in the point spectrum of the atomic Hamiltonian the physical eigenfunction _E, which has exponential decay as R , is representable by Fock's expansion.     

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

A study of the two‐dimensional hydrogen atom confined within a circle of impenetrable walls is presented. The potential inside the box is Coulomb type, whereas outside it is infinite. The energy eigenvalues and some radial wave function properties are computed with high accuracy for different box sizes. We derive the polarizability in the Kirkwood approximation, calculate the Fermi contact term as a function of the confinement radius, and investigate the filling order of the one‐electron states. When the electronic configuration of many electrons is constructed by means of the Aufbau principle, the model predicts the inversion 2s–3d levels in the N atom. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Nonunique solutions of kinetic equations

Two very hard particle models are solved and the nonuniqueness of the initial value problem for these (model) kinetic equations is explicitly demonstrated, when distribution functions decaying sufficiently slowly are permitted. The intimate connection between nonuniqueness and violation of conservation laws is made evident. The associated eigenvalue problems are solved. Finally, the general implications of these results for kinetic equations with transition rates that are increasing functions of the state variable, are stated in the form of a number of conjectures. They affect the solution of the Boltzmann equation for realistic intermolecular interactions when the collision rategI(g, ) is an increasing function of the relative velocityg.     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.     

Data analysis and representation on a general domain using eigenfunctions of Laplacian

We propose a new method to analyze and efficiently represent data recorded on a domain of general shape in by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the Karhunen–Loève transform/principal component analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.     

Eigenfunctions of the Laplacian on rotationally symmetric manifolds

Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold are constructed explicitly under the assumption that an integral of converges. This integral is the same one which gives the existence of nonconstant harmonic functions on

     

The Spectrum of Ground-State Reversed Transition of CH3OH Far-Infrared Laser

Using the CH₃OH molecular energy levels data base management program, we deduced that when a CH₃OH laser pumped by CO₂-9P(16) line, there should be a FIR laser line of 918m wavelength, which corresponding to a transition in ground-state reversed three-level energy system. For the first time, the spectra of ground-state reversed three-level system transitions were calculated by solving the density matrix equations, and the spectrum characteristics of 918m line were studied. Experimentally, the CH₃OH FIR laser line of 918m wavelength was obtained by pumping with a TEA-CO₂ laser. The experimental results were in good agreement with the theoretical calculations.     

On the ground-state threshold in random two-dimensional Ising ±J models

For square, triangular, and for hexagonal lattices there is numerical and theoretical support that the ground-state thresholdp _c between ferro- and paramagnetism in random 2D Ising ±J models, withp as the concentration of antiferromagnetic bonds, is identical top ^*which is characterized by minimal matching properties of frustrated plaquettes. From square lattices of size 100×100 we have got p_c,sq<0.117 by simulations which produced average groundstate magnetizations per spin by means of exact minimal matchings. Moreover, from the squareL×L-lattices treated (L = 10, 20, 50, 100) we obtained the estimatep _c,sq 0.1 which is in agreement with the Grinstein estimatep _c,sq 0.099 andp _c,sq 0.105 by Freund and Grassberger.     

计算双电子原子基态能量的坐标张弛变分法

给出了一种计算双电子原子基态能量和波函数的坐标张弛的变分方法.同时,利用Matlab语言开发了一个软件程序,对He原子和类He离子的基态能量进行了变分计算.     

Effect of Parabolic Potential on Bipolaron

We apply a Feynman path-integral variational approach combining with the average for the relative motion to study the stability of bipolaron in a quantum dot. The binding energy is calculated in different parameters. We find that an optimum quantum potential favors the formation of bipolaron. Compared with other methods in literature, the present approach is better than Laudau Pekar one in all coupling regime and full path-integral one in the strong coupling regime.