Gitlab public search API projects search by user

I’m looking for a way to search for public projects owned by a specific username.
There is the Projects API that can be used only if I have a GitLab user id, so this is not relevant in this case.

There is the GitLab Search API projects scope, but it seems that the following call doesn’t yield any results:<user_name>

The user i’m trying to search for does have public projects, and if I switch the <user_name> with the name of the project, it is properly returned.

Is there a specific syntax that should be used in this API to get results via username?