On the free Lévy measure of the normal distribution

Belinschi et al. [Adv. Math., 226 (2011), 3677–3698] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free Lévy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019), 1758–1787] that the normal distribution is freely selfdecomposable.