\chapter{Introduction} \label{chapter-introduction}



The Fourier transform plays an integral role in mathematics and science. It serves as the cornerstone of abstract harmonic analysis and signal processing. It is quite appealing for many reasons, namely it is a unitary operator on $L^2(\R)$ that is closely connected to the group structure of the real line and converts differential operators into multiplication operators. The Fourier transform is extremely important in quantum mechanics. In the context of quantum mechanics, the Fourier transform takes a wavefunction $\psi$ expressed in the spatial domain and then expresses it in the momentum domain and vice versa. Physically, it can be thought of as extracting the momentum information of the wavefunction. The Fourier transform also gives a rigorous mathematical justification for the Heisenberg uncertainty principle.

The Fourier transform is also critical in the spectral theory of the (negative) Laplacian: $-\Delta$. When viewed on an appropriate dense subspace of $L^2(\R)$, this operator is unbounded, essentially self-adjoint, and positive. The spectral theorem for unbounded operators says that a unitary operator $\mathcal{U}$ exists such that $-\Delta$ is unitarily equivalent to multiplication by a nonnegative function under $\mathcal{U}$. Analysis shows that $\mathcal{U}$ is exactly the Fourier transform. In one real variable, $-\frac{d^2}{dx^2}$ has an image of $y^2$ under the Fourier transform. (In higher dimensions, $-\sum_i \frac{d^2}{dx_i^2}$ has image $\sum_i x_i^2$.) Likewise, the multiplication operator $x^2$ has an image of $-\frac{d^2}{dy^2}$ under the Fourier transform. As such, the operator $-\frac{d^2}{dx^2} + x^2$ will be form-invariant under the Fourier transform. This can be readily realized as the quantum-mechanical harmonic oscillator Hamiltonian.

The purpose of this thesis is to explore integral transforms which generalize the Fourier transform and develop Hamiltonian-like objects which generalize the quantum-mechanical harmonic oscillator Hamiltonian. This pursuit leads to a rich, general theory when coupled with the language of supersymmetric quantum mechanics. This new theory is denoted as coupled supersymmetry, or coupled SUSY. In this framework, one can define pairs of integral transforms that generalize the Fourier transform which are closely related. In the context of the quantum-mechanical harmonic oscillator, these structures collapse into pairs of duplicates, and so these structures are hidden in this case.

Generalizing the Fourier transform on $\R$ and the quantum-mechanical harmonic oscillator are hardly new ideas. The Hankel, or Fourier-Bessel, transform \cite{hankel,jaming,grafakos} and the fractional Fourier transform \cite{namias} are two such generalizations of the Fourier transform. The Fourier-Bessel transform is very closely related to the work contained herein. There has been some work characterizing and studying general integral transforms on $\R$ \cite{chen,ma} and some work on periodic integral transforms \cite{walters,mccallum}. There is also a generalization of the Fourier transform to arbitrary locally compact abelian groups which subsumes both Fourier series and Fourier transforms on $\R$. This generalization of the Fourier transform is quite general and very satisfactory, though it does not in general capture a differential structure. This generalization is founded on continuous homomorphisms of the locally compact abelian group to the unit circle $\mathbb{T}$. Such continuous homomorphisms serve as the integral kernel for the Fourier transform. It is a simple exercise to show that in the case of the group of real numbers under addition, the Fourier kernel is effectively unique and is exactly $\exp(-ixy)$. Furthermore, Fourier transforms come in pairs in this setting: one may define a Fourier transform on a group $G$ and its dual group $\widehat{G}$. A generalization closely related to the abstract harmonic analytic Fourier transform is the Dunkl transform \cite{jeu,Rösler}.

Supersymmetric quantum mechanics is a broad generalization of the ladder operator formalism of the quantum-mechanical harmonic oscillator. The connection between the quantum-mechanical harmonic oscillator and the Fourier transform can be made explicit beyond the connection via the Gaussian as an eigenfunction of each. The creation and annihilation operators $x\pm ip$ in the quantum-mechanical harmonic oscillator are \emph{eigenoperators} \cite{mccallum} of the Fourier transform, i.e., they are form-invariant under the Fourier transform. This parallel between the quantum-mechanical harmonic oscillator and the Fourier transform suggests that, at least in special subcases of supersymmetric quantum mechanics, Fourier-like transforms exist and have the supersymmetric charge operators as eigenoperators.

This thesis has two main sections. The first section comprises Chapters 2 and 3 and focuses on integral transform theory. Chapter 2 contains a brief overview of Fourier series and transform theory. Chapter 3 contains generalizations of the Fourier transform and explores uncertainty relations and a notion of duality between integral transforms. The analysis of the generalized Fourier transform leads to the consideration of Hamiltonian-like operators which have a supersymmetric nature. This leads naturally into the second section of the thesis. The second section comprises Chapters 4 and 5. Chapter 4 contains a brief review of the quantum-mechanical harmonic oscillator and supersymmetric quantum mechanics. Chapter 5 introduces coupled supersymmetry and explores its analytic and algebraic structures.

The work in Chapter 3 has been accepted to the \emph{Journal of Fourier Analysis and Applications} \cite{williams1} and is to appear. The work in Chapter 5 has been submitted to arXiv \cite{williams2}.





