We study unimodality for free multiplicative convolution with free normal
distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on
unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal
distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time
$t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on
$\{e^{i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in
(0,\pi/2)$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large
$t>0$; (3) ${\bf b} \boxtimes \lambda_t$ is not unimodal at any time $t>0$,
where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$;
(4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$.
Moreover, we study unimodality for classical multiplicative convolution (with
Poisson kernels), which is useful in proving the above four results.Comment: 19 pages, 4 figure