In this thesis, we will show how certain classes of quantum many-body Hamiltonians with $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ 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 $k\times k$ matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions. The SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have $\su{2}$ spectrum generating algebras. Next, we extend the technique to the non-compact $\su{1,1}$ 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 $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebra. An 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 $10\times 10$ 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. The thesis concludes by exploring possible future developments of the SHA.

In this thesis, we will show how certain classes of quantum many-body Hamiltonians with $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ 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 $k\times k$ matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions. The SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have $\su{2}$ spectrum generating algebras. Next, we extend the technique to the non-compact $\su{1,1}$ 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 $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebra. An 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 $10\times 10$ 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. The thesis concludes by exploring possible future developments of the SHA.