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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 ofeigenvectors form an orthonormal basis of $L^2(\mathbb{R}^+).$

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