On the Pseudo-Differential Operator (−x−1D)ν

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]