From Generalized Fourier Transforms to Coupled Supersymmetry

The Fourier transform, the quantum mechanical harmonic oscillator, and supersymmetric quantum mechanics are well-studied objects in mathematics. The relations between them are also well-understood, though the traditional understanding of each has led to a rigidity in their design. In this thesis, we will develop generalizations of the Fourier transform and explore their relations. Exploring the analytic structure of these integral transforms, particularly their related Hamiltonians, naturally leads to Hamiltonian-like operators which have rich analytic and algebraic structures. The algebraic structure underlying the Hamiltonian-like operators associated to generalized Fourier transforms suggests a much-more-general abstract formulation. To this end, we introduce the coupled supersymmetry (coupled SUSY) algebraic framework which unifies the quantum mechanical harmonic oscillator and supersymmetric quantum mechanics in a more complete way. The coupled SUSY framework subsumes the quantum mechanical harmonic oscillator and provides a broader class of systems which have rich functional analytic and algebraic structures. In this setting, one is able to develop further generalizations of the Fourier transform. A further generalization of coupled SUSY is briefly presented which appears to be a very new insight into supersymmetry.