On eigenproblem for inverted harmonic oscillators

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\mathrm{\infty }}(\mathbb{R})$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is $\mathbb{C}$. The spectrum of the differential operator $-\frac{d}{d{x}^2}-{\omega }^2{x}^2$ is continuous and has physical significance only for the states which are in $L^2(\mathbb{R})$ and correspond to real eigenvalues. To identify them we use two approaches. First we define a unitary operator between $L^2(\mathbb{R})$ and $L^2$ for two copies of $\mathbb{R}$. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator $-i\frac{d}{dx}$. This shows that the (generalized) spectrum of the inverted harmonic operator is real. The second approach uses rigged Hilbert spaces.