Whoops you re right I was looking closely Your T is the equivalent to the operator for the 1d quantum harmonic oscillator which means that its eigenfunctions are Hermite functions which are the same as the eigenfunctions of the Fourier transform and the eigenvalues are right too up to a scale factor which I haven t looked closely enough to check So yes it does indeed look like this might give the FRFT This does seem like a nice way of doing it If it works like it appears to I would be surprised if it weren t known since the FRFT is often defined precisely by looking at Hermite functions and the connection of Hermite functions to the harmonic oscillator eigenproblem is well known However we would have to dig up a clear reference before we could put it into the article