It's always a good idea to send an invoice as ebays emails are only confirmations.

I send an invoice immediately and ebay send an automatic reminder 2 days after purchase.

I usually send a message at 5 days asking if there is a problem with payment.

If no response, I open an Unpaid Item Case and that can sometimes spur a response or payment.
If not, I close the case as soon as I can, 4 days after it's opened.
Final fees are returned, buyer gets a strike, I add them to my Block Bidders List and re-list the item.

In this caseI would open a case and not wait a month, unless the buyer contacts you personally and makes arragements.

They obviously have access to their account to make the purchase.