Supersymmetric Resolvent-Based Fourier Transform

Abstract

We calculate in a numerically friendly way the Fourier transform of a non-integrable function, such as , by replacing F with R -1 FR, where R represents the resolvent for harmonic oscillator Hamiltonian. As contrasted with the non-analyticity of at in the case of a simple replacement of F by , where and represent the momentum and position operators, respectively, the turns out to be an entire function. In calculating the resolvent kernel, the sampling theorem is of great use. The resolvent based Fourier transform can be made supersymmetric (SUSY), which not only makes manifest the usefulness of the even-odd decomposition of in a more natural way, but also leads to a natural definition of SUSY Fourier transform through the commutativity with the SUSY resolvent.