arXiv subjects
Publications and source records attributed to Miloslav Havlíček.
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 $L^2(\mathbb{R}^+).$
The paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is defined and applied to finite-dimensional representations of $sl(n,\mathbb{C})$ in relation with its $\mathbb{Z}_2$-gradings. For representation theory of $sl(n,\mathbb{C})$ the Gel'fand-Tseitlin method turned out very effective.