Goce Chadzitaskos
Parabolic cylinder functions as orthonormal bases on $L^2(\mathbb R^+)$ and $L^2(\mathbb R)$
In addition to orthogonal polynomials, orthogonal functions also play an important role and have a wide range of uses. They are related to the solution of differential equations. In this contribution we present the explicit form of one parameter families of orthonormal bases on spaces $L^2(\mathbb R^+)$ and $L^2(\mathbb R).$ The bases are formed by eigenvectors of the self-adjoint extension of Schr\"odinger operator of the asymmetric harmonic oscillator. For each parameter the set of eigenvectors form an orthonormal basis on $L^2(\mathbb R^+)$ or $L^2(\mathbb R)$. The Hermite polynomials are done by special parameters.
