The seniority-zero excited-state spectrum of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}+{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}k})\ensuremath{-}G \ensuremath{\Sigma}\stackrel{}{k,l>0}{{a}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\ensuremath{-}l}{a}_{l}$ is studied in detail. Approximate excited-state wave functions of the form ${\ensuremath{\psi}}^{\ensuremath{\lambda}}=\ensuremath{\Sigma}\stackrel{}{\ensuremath{\alpha}}F(\ensuremath{\lambda},\ensuremath{\alpha})\ensuremath{\Sigma}\stackrel{}{i}D(\ensuremath{\alpha},i){{a}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Sigma}\stackrel{}{j>i}D(\ensuremath{\alpha},j){{a}_{j}}^{\ifmmode\dagger\else\textdagger\fi{}}{{a}_{\ensuremath{-}j}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdots}|0〉$ are developed. These solutions are compared with exact solutions of a small system and the agreement is quite good. A somewhat larger system is studied in order to see how the excited-state spectrum changes as the number of separable functions is increased. The method is applied to heavy nuclei and is in good agreement with observed
0+ excited states for nuclei having 144 to 150 neutrons. A theoretical
0+ spectrum is displayed for each even neutron system from 144 to 152 neutrons.
The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.
In this thesis, we will show how certain classes of quantum many-body Hamiltonians with
su21oplussu22oplusldotsoplussu2k spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on
k. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of
k equations and diagonalizing
ktimesk matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have
su2 spectrum generating algebras. Next, we extend the technique to the non-compact
su1,1 algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a
k-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a
su21oplussu22oplusldotsoplussu2k spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of
10times10 matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.