Orthonormal bases on $L^2(\mathbb{R}^+)$

We derive the explicit form of eigenvectors of selfadjoint extension $H_\xi$, parametrized by $\xi \in \langle 0,\pi),$ of differential expression $H=-\frac{d^2 }{d x^2} + \frac{x^2 }{4}$ together with the spectrum $\sigma(H_\xi)$ on the space $L^2(\mathbb{R}^+).$ For each $\xi$ the set of eigenvectors form an orthonormal basis of $L^2(\mathbb{R}^+).$