Exact and approximated solutions for the harmonic and anharmonic repulsive oscillators: Matrix method

We study the repulsive harmonic oscillator and an extension of this system, with an additional linear anharmonicity term. The system is solved in exact and perturbative form, the latter by using the so-called normalized perturbative matrix method. The perturbative solution up to second order is compared with the exact solution when the system is initially in coherent and a Schrödinger-cat states