abstract
- We present two equivalent analytical expressions of the Wigner function (WF) of Ince-Gauss (IG) eigenstates of the Schrödinger equation in elliptic coordinates for the two-dimensional (2D) harmonic oscillator. To this end, we first derive closed-form expressions for the Weyl transforms for two different Hermite-Gauss (HG) states and two different helical Laguerre-Gauss (LG) states. Using these Weyl transforms, we also obtain the WF of the IG states through twofold finite summations and the WF of parity-definite LG states in a closed form. An alternative expression of the WF of the IG states in terms of the harmonic oscillator's quadratic invariants is also presented. © 2025 The Author(s).