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Abstract

We consider an eigenvalue problem for an inverted one-dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty }(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is ${\mathbb C}$. The spectrum of the differential operator $-{\frac {d}{dx^2}}-{\omega }^{2}{x^2}$ 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 orthonormalize in Dirac sense the states corresponding to real eigenvalues. This leads to the doubly degenerated real line as the spectrum of the Hamiltonian (in $L^2({\mathbb R})$). We also use two other 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 again that the (generalized) spectrum of the inverted harmonic operator is a doubly degenerated real line. The second approach uses rigged Hilbert spaces.