On the pseudo-differential operator (-x-1D)ν

Abstract

For a certain Frechet space F consisting of complex-valued C∞ even functions defined on R and rapidly decreasing as |x| → ∞, we show that if ν is any complex number, • The pseudo-differential operator (-) is an automorphism on . • Re α > 0, is an eigenfunction of the pseudo-differential operator (-). • For in, a linear subsapce of the Hilbert space generated by the even-order Hermite functions = 0, 1, 1, [formula] where C2k and an-j are constants and [formula].