My research has two main components lying in the field of functional analysis: integral transform theory and coupled supersymmetry and their connections.

My work in integral transform theory is concerned with real line generalizations of the Fourier transform, specifically the development of unitary integral transforms. I have worked on the Φ transform, a full-line generalization of the Fourier transform that shares many properties with the Fourier transform. My current work in this direction is on the Fourier–Bessel transform, a generalization of the Fourier cosine transform with a Bessel function appearing in the integral kernel. Its natural domain is on the half-line restriction of even functions. The Fourier–Bessel transform has numerous desirable properties mirroring those of the Fourier transform. Currently, I am preparing a comprehensive survey article on the Fourier–Bessel transform for newcomers and experts alike. The Fourier–Bessel transform has attracted numerous generalizations and partners, including Fourier sine transform generalizations. I am currently working on a natural Fourier sine transform generalization that acts as a partner transform to the Fourier–Bessel transform.

Coupled supersymmetry (or coupled SUSY for short) is a specific flavor of operator theory that finds a happy medium between the quantum harmonic oscillator and supersymmetric quantum mechanics. The quantum harmonic oscillator serves as the foundation for much of high energy physics due to its nice Lie algebra structure: it has ladder operators (creation/raising and annihilation/lowering operators) that allow one to traverse the entire spectrum from a single eigenstate (typically the Gaussian ground state). Supersymmetric quantum mechanics attempts to generalize the algebraic properties of the quantum harmonic oscillator to more general quantum systems, though most of these lack enough structure to be able to determine the spectra of the associated Hamiltonians (with exception to the shape-invariant potentials). Indeed, supersymmetric quantum mechanics only has a graded Lie algebra structure that allows transferring between two different Hilbert spaces but does not allow one to traverse up or down the spectral ladders. Coupled SUSY addresses this by adding further constraint between the Hamiltonians, allowing for exact knowledge of eigenvalues and eigenfunctions.

While these two avenues of research seem at odds, there is a lot of interplay between them. In fact, coupled SUSY originally sprung out of determining the eigenfunctions of the Φ transform. It was found that its eigenfunctions were also eigenfunctions of a Hamiltonian resembling the quantum harmonic oscillator with a major exception: there were multiple potential factorizations which were closely linked, resulting in coupled SUSY. Moreover, the Fourier transform as an abstract operator acting on the Hermite-Gauss basis (the basis of eigenfunctions of the quantum harmonic oscillator) is equivalent to an exponentiation of the harmonic oscillator Hamiltonian. They are also connected through the uncertainty principle: often, uncertainty principles for integral transforms can be recast as eigenvalue problems for Hamiltonians via the calculus of variations.