Let  $S$ be a  nonnegative self- adjoint operator in a  complex Hilbert Space $X$ (say $X$ consists of functions on $\mathbb R^d$, e.g., $L^2(\mathbb R^d), \dot{H}^1(\mathbb R^d)$, etc.. )

We consider abstract Cauchy problem  for the Schr\"odinger equation (SE):

$$i\frac{\partial}{\partial t} u(x,t) + Su(x,t)=0, u(x,0)=u_0(x)$$

Formally, we  may write the solution of $(SE)$ as
$$u(x,t)= e^{itS}u_0(x).$$

>My Question: If $u_0(x)\in X$ is [radial](https://en.wikipedia.org/wiki/Radial_function), then can we expect $e^{itS}u_0(x)$ also a radial?

Note: (1) I know if $S=- \Delta$, then $e^{itS}u_{0}$ is radial  whenever $u_0$ is radial. (2)  Specifically, I am interested in $S=-\Delta+ \frac{a}{|x|^2}$, see here [Section 1.1](https://arxiv.org/abs/1503.02716) for detail (with $d\geq 3, a\geq (\frac{d-2}{2})^2$)