# Bargmann-type transforms and modified harmonic oscillators

## TL;DR

This paper explores Bargmann-type transforms and their associated eigenfunctions for harmonic oscillators, analyzing both commutative and non-commutative cases, and introduces a complete orthogonal system related to phase plane ellipses.

## Contribution

It introduces a new class of orthonormal systems derived from Bargmann-type transforms and provides explicit eigenvalues and eigenfunctions for certain non-commutative harmonic oscillators.

## Key findings

- Eigenfunctions form complete orthonormal systems
- Explicit eigenvalues for commutative non-commutative harmonic oscillators
- Complete orthogonal system related to phase plane ellipses

## Abstract

We study some complete orthonormal systems on the real-line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real-line.