I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have$$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$where $H = -\frac{d^2}{dx^2} + x^2$ is the quantum harmonic oscillator or is this false?
It feels natural since $ -\frac{d^2}{dx^2} \le H$ in the sense of operators.
