A Swanson-like Hamiltonian and the inverted harmonic oscillator

Abstract

We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian H ( theta ) which is closely related to the Swanson Hamiltonian, and we construct bi-coherent states for it. After that, we show how and in which sense the eigensystem of the Hamiltonian H of the inverted quantum harmonic oscillator can be deduced from that of H ( theta ). We show that there is no need to introduce a different scalar product using some ad hoc metric operator, as suggested by other authors. Indeed we prove that a distributional approach is sufficient to deal with the Hamiltonian H of the inverted oscillator.