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

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

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