Improved strategies for variational calculations for helium

The aim of this work is to apply trial functions constructed from Hylleraas functions with three independent sets of nonlinear scale factors to variational calculations for helium and helium-like ions. The ground state and low-lying Rydberg energy levels of these ions have been calculated to several orders of magnitude greater accuracy than previous work in this area while using an equal, or in most cases, a reduced number of basis functions. Each of the three sectors of the basis set is found to describe a different scale of coordinate space corresponding to the asymptotic, intermediate, and close-ranged distances between particles. The incorporation of the third, close-ranged sector, allows the basis set to better model complex correlation effects between the nucleus and the two electrons in the atomic three-body problem. Optimization of the basis set parameters is achieved through standard variational techniques and the validity of the wave functions near the electron-nucleus and electron-electron coalescence points is tested using the Kato cusp conditions. The tripled basis set is also applied to the 1/ Z perturbation expansion as a case study. A multiple-precision package, MPFUN90 written by David H. Bailey, was used to alleviate numerical instabilities which arose for certain states.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2004 .N57. Source: Masters Abstracts International, Volume: 43-01, page: 0225. Adviser: G. W. F. Drake. Thesis (M.Sc.)--University of Windsor (Canada), 2004

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $\Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P \in \psi^m(M; E_0, E_1)$ be a $\Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $\Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $\alpha$ be an irreducible representation of the group $\Gamma$. Then $P$ induces by restriction a map $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{s-m}(M; E_1)_\alpha$ between the $\alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $\pi_\alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $\Gamma$ is finite abelian. We prove then that the restriction $\pi_\alpha(P)$ is Fredholm if, and only if, $P$ is "$\alpha$-elliptic", a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $\alpha$-elliptic, but the converse is not true in general. However, if $\Gamma$ acts freely on a dense open subset of $M$, then $P$ is $\alpha$-elliptic for the given fixed $\alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $\psi^{m}(M; E)^\Gamma$ of classical, $\Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E \to M$ and of the structure of its restrictions to the isotypical components of $\Gamma$. These structures are described in terms of the isotropy groups of the action of the group $\Gamma$ on $E \to M$

Ground-state energies for helium, H-, and Ps-

Nonrelativistic energy and other properties of He, H- and Ps- were discussed using a triple basis set in Hylleraas coordinates. The stability and efficiency of the computational method was compared with the quasirandom method. Results showed that the triple basis set in Hylleraas coordinates is capable of exceeding the accuracy of calculations for three-body system based on quasirandom Monte Carlo methods

Groupoids and an index theorem for conical pseudo-manifolds

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds

Muon capture on light nuclei

This work investigates the muon capture reactions 2H(\mu^-,\nu_\mu)nn and 3He(\mu^-,\nu_\mu)3H and the contribution to their total capture rates arising from the axial two-body currents obtained imposing the partially-conserved-axial-current (PCAC) hypothesis. The initial and final A=2 and 3 nuclear wave functions are obtained from the Argonne v_{18} two-nucleon potential, in combination with the Urbana IX three-nucleon potential in the case of A=3. The weak current consists of vector and axial components derived in chiral effective field theory. The low-energy constant entering the vector (axial) component is determined by reproducting the isovector combination of the trinucleon magnetic moment (Gamow-Teller matrix element of tritium beta-decay). The total capture rates are 393.1(8) s^{-1} for A=2 and 1488(9) s^{-1} for A=3, where the uncertainties arise from the adopted fitting procedure.Comment: 6 pages, submitted to Few-Body Sys

Elliptic operators on manifolds with singularities and K-homology

It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula for K-groups of algebras of pseudodifferential operators.Comment: revised version; 25 pages; section with applications expande

Self-trapped states and the related luminescence in PbCl2_2 crystals

We have comprehensively investigated localized states of photoinduced electron-hole pairs with electron-spin-resonance technique and photoluminescence (PL) in a wide temperature range of 5-200 K. At low temperatures below 70 K, holes localize on Pb$^{2+}$ ions and form self-trapping hole centers of Pb$^{3+}$. The holes transfer to other trapping centers above 70 K. On the other hand, electrons localize on two Pb$^{2+}$ ions at higher than 50 K and form self-trapping electron centers of Pb$_2$$^{3+}$. From the thermal stability of the localized states and PL, we clarify that blue-green PL band at 2.50 eV is closely related to the self-trapped holes.Comment: 8 pages (10 figures), ReVTEX; removal of one figure, Fig. 3 in the version