Resolvent of Harmonic Oscillator Hamiltonian and Its Application to Fourier Transform for Generalized Functions

摘要

For the Fourier transformFof a non-integrablefunction Φ, we exploit theresolventRforthe harmonic oscillator Hamiltonian, where the integral kernel for R can be represented using the confluent hypergeometric function. Due to the commutativity of F and R, F can be regarded by R~(-1)FR. In the case ofΦ(x)= 1, for example, it follows that(RΦ)(x) is continuous on R and that (RΦ)(x) ～ x~(-2)(|x| → ∞), so that RΦ turns outto be integrable over R. The finding that(FR)Φis exponentially localized indicatesthat the mapFR: Φ→ ф can be used as data compression ofΦ. Moreover, the inverse mapR~(-1)F~(-1): ф→ Φ is well defined, which implies that the data decompression into Φ can be made in a numerical calculation friendly way.