How to use siteSearch and siteSearchFilter in Google Custom Search JSON/Atom API (v1)?

Question

Google

I'm writing some stuff that will need to use the Google Custom Search JSON/Atom API (v1).
There is one optional but very important siteSearch parameter for me, which indicates the address of the site on which you want to search (similar to the site: parameter in a regular Google search). And so this parameter somehow strangely works.
For example siteSearch=habrahabra.ru; q=php (analogous to "site:habrahabr.ru php" of Google search) gives 46,500 results against 182,000 with a regular Google search.
And if I do a search on a small, but also indexed site, I generally get 0 results in the GCS API, despite the fact that in a regular Google search for this site there are a couple of hundred results.

Who worked with this API is there an opportunity to overcome this problem?