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

TL;DR

This paper provides exact solutions for the energies and wave functions of two-mode squeezed harmonic oscillators and analyzes the mathematical well-definedness of higher-order harmonic generation within Bargmann-Hilbert spaces.

Contribution

It derives closed-form solutions for certain squeezed oscillators and clarifies the ill-defined nature of higher-order harmonic generation in this mathematical framework.

Findings

Exact energy and wave function expressions for displaced and squeezed oscillators.

Higher-order harmonic generation ($k extgreater 2$) lacks eigenfunctions in Bargmann-Hilbert space.

Higher-order harmonic generation is ill-defined in the studied framework.

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.