Solving two-mode squeezed harmonic oscillator and $k$th-order harmonic generation in Bargmann-Hilbert spaces

Abstract

We analyze the two-mode squeezed harmonic oscillator and the $k$th-order harmonic generation within the framework of Bargmann-Hilbert spaces of entire functions. For the displaced, single-mode squeezed and two-mode squeezed harmonic oscillators, we derive the exact, closed-form expressions for their energies and wave functions. For the $k$th-order harmonic generation with $k\geq 3$, our result indicates that it does not have eigenfunctions and is thus ill-defined in the Bargmann-Hilbert space.