Whoops youre right I was looking closely  Your T is the equivalent to the operator for the d 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 havent 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 werent known since the FRFT is often defined precisely by looking at Hermite functions and the connection of Hermite functions to the harmonicoscillator eigenproblem is well known  However we would have to dig up a clear reference before we could put it into the article