Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

The goal of this dissertation has been to develop a method that enables one to calculate accurate, rigorous lower bounds to the eigenvalues of the standard nonrelativistic spin-free Hamiltonian for an atom with N electrons. Lower bounds are necessary in order to complement upper bounds obtained from the Hartree-Fock and Rayleigh-Ritz techniques. Without accurate lower bounds, it is impossible to estimate the error of the approximate values of the energies. By combining two heretofore distinct methods and using the symmetry properties of the Hamiltonian, this goal has been achieved. The first of the two methods is the method of intermediate problems. By beginning with an appropriately chosen "base operator" H⁰, one forms a sequence of intermediate Hamiltonians Hᵏ, k = 1,2,..., whose corresponding eigenvalues form a sequence of lower bounds to the eigenvalues of the original Hamiltonian H. Complications which occurred in this method due to the stability of essential spectra under compact perturbations were later surmounted with the use of abstract separation of variables by D. W. Fox. The second technique, the effective field method, provides a lower bound operator to the interelectron repulsion term in H that is of the form of a sum of separable potentials. This latter technique reduces the eigenvalue problem for H⁰ to a sum of single particle operators. With the use of a special potential, the Hulthen potential, one may construct an explicitly solvable base problem from the effective field method, if one uses the method of intermediate problems to calculate lower bounds to non-S states. This base problem is then suitable as a starting point for the method of intermediate problems with the Fox modifications. The eigenvalues of the new base problem are already comparable to the celebrated Thomas-Fermi energies. The final part of the dissertation provides a practical procedure for determining the physically realizable spectra of the intermediate operators. This is accomplished by restricting the Hamiltonian to subspaces of proper physical symmetry so that the resulting lower bounds will be to eigenvalues of physical significance.

Chee Shiu Chung = 阻尼振子的線空間結構與哈密頓理論 / 朱兆中. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 88). / Text in English; abstracts in English and Chinese. / Chee Shiu Chung = Zu ni zhen zi de xian kong jian jie gou yu ha mi dun li lun / Zhu Zhaozhong. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Conservative Systems --- p.4 / Chapter 2.1 --- General Formalism --- p.4 / Chapter 2.2 --- One Simple Harmonic Oscillator --- p.7 / Chapter 2.3 --- Two Coupled Harmonic Oscillators --- p.9 / Chapter 3 --- Dissipative Systems --- p.12 / Chapter 3.1 --- Elimination of Bath --- p.12 / Chapter 3.2 --- One Oscillator with Dissipation --- p.16 / Chapter 3.3 --- Two Oscillators with Dissipation --- p.19 / Chapter 4 --- Eigenvector Expansion and Bilinear Map --- p.21 / Chapter 4.1 --- Formalism --- p.21 / Chapter 4.2 --- Inner Product and Bilinear Map --- p.23 / Chapter 4.3 --- Normalization and Phase --- p.25 / Chapter 4.4 --- Matrix Representation --- p.25 / Chapter 4.5 --- Duality --- p.28 / Chapter 5 --- Applications and Examples of Eigenvector Expansion --- p.31 / Chapter 5.1 --- Single Oscillator --- p.31 / Chapter 5.2 --- Two Oscillators --- p.32 / Chapter 5.3 --- Uneven Damping --- p.33 / Chapter 6 --- Time Evolution --- p.36 / Chapter 6.1 --- Initial-Value Problem --- p.36 / Chapter 6.1.1 --- Green's Function --- p.37 / Chapter 6.2 --- Sum Rules --- p.39 / Chapter 7 --- Time-Independent Perturbation Theory --- p.41 / Chapter 7.1 --- Non-degenerate Perturbation --- p.41 / Chapter 7.2 --- Degenerate Perturbation Theory --- p.46 / Chapter 8 --- Jordan Block --- p.48 / Chapter 8.1 --- Jordan Normal Basis --- p.48 / Chapter 8.1.1 --- Construction of Basis Vectors --- p.48 / Chapter 8.1.2 --- Bilinear Map --- p.50 / Chapter 8.1.3 --- Example of Jordan Normal Basis --- p.55 / Chapter 8.2 --- Time Evolution --- p.56 / Chapter 8.2.1 --- Time Dependence of Basis Vectors --- p.56 / Chapter 8.2.2 --- Initial-Value Problem --- p.58 / Chapter 8.2.3 --- Green's Function --- p.59 / Chapter 8.2.4 --- Sum Rules --- p.60 / Chapter 8.3 --- Jordan Block Perturbation Theory --- p.61 / Chapter 8.3.1 --- Lowest Order Perturbation --- p.61 / Chapter 8.3.2 --- Higher-Order Perturbation --- p.65 / Chapter 8.3.3 --- Non-generic Perturbations --- p.66 / Chapter 8.4 --- Examples of High-Order Criticality --- p.66 / Chapter 8.4.1 --- Fourth-order JB --- p.67 / Chapter 8.4.2 --- Third-order JB --- p.74 / Chapter 8.4.3 --- Two Second-order JB --- p.79 / Chapter 9 --- Conclusion --- p.81 / Chapter A --- Appendix --- p.83 / Chapter A.l --- Fourier Transform and Contour Integration --- p.83 / Chapter B --- Degeneracy and Criticality --- p.86 / Bibliography --- p.88

Graduation date: 2003

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

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We study the Hamiltonian structure of an important class of nonlinear partial differential equations: the so-called systems of hydrodynamic type, which are first-order in tempo-spatial variables, and quasi-linear. Chapters 1 and 2 constitute a review of background material, while Chapters 3, 4, 5 contain new results, with additional review sections as necessary. In Chapter 3 we demonstrate, via the Nijenhuis tensor, the integrability of a system of hydrodynamic type derived from the classical Volterra system. In Chapter 4, families of Hamiltonian structures of hydrodynamic type are constructed, as well as a gauge transform acting on Hamiltonian structures of hydrodynamic type. In Chapter 5, we present necessary and sufficient criteria for a three-component system of hydrodynamic type to be Hamiltonian, and classify the Lie-algebraic structures induced by a Hamiltonian structure for four-component systems of hydrodynamic type. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-12-23 11:35:41.976