An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian

We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of $\mathfrak{s}\mathfrak{u}(1,1)$, and of a generalized version of it relevant in presence of non Hermitian generators.