You have to go to the Child Support Agency in the USA!!! They will then use the REMO agreement between countries to get him to pay the maintenance due.
The proceedings, the amount to be paid, etc ... will be calculated in the USA, not in Switzerland. The Swiss Government will only enforce the payments!!
I am doing the same, but I live in Switzerland and the father of my child lives in the UK. So, I went to the Child Support here in Switzerland.
It's very hard time and it will probably take very long, but I am hanging in there!!
Good luck!!!
JO