## A generalization of the displaced parity

A generalization of the displaced parity operator SB-222200 the following operator which we call bifractional displacement operator. The bifractional displacement operator is a unitary operator and defined asequation(7)U(α,β;θα,θβ)= cos?(θα−θβ) 1/2∫dα′dβ′Δ(β,α′;θβ)Δ(α,−β′;θα)D(α′,β′)U(α,β;θα,θβ)= cos?(θα−θβ) 1/2∫dα′dβ′Δ(β,α′;θβ)Δ(α,−β′;θα)D(α′,β′)[U(α,β;θα,θβ)]†=U(−α,−β;−θα,−θβ)[U(α,β;θα,θβ)]†=U(−α,−β;−θα,−θβ)

Here we replaced the two Fourier transforms in Eq. (6) with two fractional Fourier transforms. We note that the two fractional Fourier transforms, use the variables α′,β′α′,β′ which are related to position and momentum and are dual to each other. In lysosomes sense our two-dimensional fractional Fourier transform is not a straightforward generalization of a one-dimensional fractional Fourier transform.