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Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 233-234).

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The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research. / Ph. D. / incomplete_metadata

A new direct full variational approach is described. The approach exploits a tensor (Kronecker) product construction of the many-electron Hamiltonian and has a number of computational advantages. Explicit assembly and storage of the Hamiltonian matrix is avoided by using the Kronecker product structure to form matrix-vector products directly from the molecular integrals. Computation-intensive integral transformations and formula tapes are unnecessary. The wavefunction is expanded in terms of spin-free primitive kets rather than Slater determinants or configuration state functions and is equivalent to a full configuration interaction expansion. The approach suggests compact storage schemes and algorithms which are naturally suited to parallel and pipelined machines. Sample calculations for small two- and four-electron systems are presented. The preliminary ground state potential energy surface of the hydrogen molecule dimer is computed by the tensor product method using a small basis set. / Ph. D.

We begin this dissertation by presenting a brief introduction to the theory of solitons and integrability (plus some classical methods applied in this field) in Chapter 1, mainly using the Korteweg-de Vries equation as a typical model. At the end of this Chapter a mathematical framework of notations and terminologies is established for the whole dissertation. In Chapter 2, we first introduce two specific matrix spectral problems (with 3 potentials) associated with matrix Lie algebras $\mbox{sl}(2;\mathbb{R})$ and $\mbox{so}(3;\mathbb{R})$, respectively; and then we engender two soliton hierarchies. The computation and analysis of their Hamiltonian structures based on the trace identity affirms that the obtained hierarchies are Liouville integrable. This chapter shows the entire process of how a soliton hierarchy is engendered by starting from a proper matrix spectral problem. In Chapter 3, at first we elucidate the Gauge equivalence among three types $u$-linear Hamiltonian operators, and construct then the corresponding B\"acklund transformations among them explicitly. Next we derive the if-and-only-if conditions under which the linear coupling of the discussed u-linear operators and matrix differential operators with constant coefficients is still Hamiltonian. Very amazingly, the derived conditions show that the resulting Hamiltonian operators is truncated only up to the 3rd differential order. Finally, a few relevant examples of integrable hierarchies are illustrated. In Chapter, 4 we first present a generalized modified Korteweg-de Vries hierarchy. Then for one of the equations in this hierarchy, we build the associated Riemann-Hilbert problems with some equivalent spectral problems. Next, computation of soliton solutions is performed by reducing the Riemann-Hilbert problems to those with identity jump matrix, i.e., those correspond to reflectionless inverse scattering problems. Finally a special reduction of the original matrix spectral problem will be briefly discussed.