Spectral analysis and integral representations of the tempered fractional Riesz derivative

In this work, we develop the harmonic analysis associated with the second-order differential operator $ \mathscr{L}_\gamma = -\frac{d^2}{dx^2} -2\gamma\, \frac{d}{dx} - \gamma^2 $. Fractional powers of $ \mathscr{L}_\gamma $ are defined via spectral representation, and a singular integral representation is provided. Furthermore, we establish the equivalence between the fractional powers of $ \mathscr{L}_\gamma $ and a tempered Riesz derivative.