UC Santa Barbara

The bootstrap program in theoretical physics arose out of attempts to determine the structure of conformal field theories in a constraint-oriented, non-Hamiltonian fashion. In recent years, bootstrap methods have seen a strong resurgence in the numerical analysis of quantum mechanical systems without conformal symmetry. In this thesis, we develop the theory of the bootstrap approach to one-dimensional quantum systems. We show how Schrodinger quantum mechanics can be numerically solved by the method and develop the algebraic theory required to adapt the approach to domains with boundary and to scattering problems. We develop and implement efficient semidefinite program- ming algorithms to rigorously and numerically determine the energy spectra of confining polynomial potentials in one dimension. We also address the problem of spin chains and demonstrate how bootstrap methods can be used to regress conformal data from numeri- cal approximations of the spin correlation functions. Broadly, this work lays a foundation for the theory and application of bootstrap methods to a wide variety of quantum mechanical systems, and also provides a mathematical background of the method and a thorough review of related work in high-energy theory, condensed matter physics, and mathematical optimization. Future directions for research both in wave mechanics and interacting spin systems are proposed.