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

Abstract

We derive the explicit form of eigenvectors of selfadjoint extension $H_{\xi }$, parametrized by $\xi \in ⟨0,\pi ),$ 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 $L^2({\mathbb{R}}^{+}).$