Answer by Christian Remling for Bound in terms of harmonic oscillator
An easy way to see that this isn't working is as follows: Take $\psi=\psi_0=e^{-x^2/2}$ as the ground state of $H$, so $H\psi=\psi$. Since $\psi''=(x^2-1)\psi$ is not a multiple of $\psi_0$, we have...
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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...
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