They can quickly get a court subpoena and have your telecom provider put a trace/trap on the phone line to find out where the calls are is coming from.
The information provided by this telecom service is typical, though some of it is specific to XFINITY/Comast.
PROVIDED AS A SAMPLE, contact your telecom provider for specific instructions
http://customer.comcast.com/help-and-support/phone/about-call-trace/
XFINITY Voice Features: Call Trace
Updated on February 23, 2015 at 6:28 PM
Introduction
Our XFINITY Voice security feature, Call Trace, protects you against harassment from another caller, by tracing their number. This information will be available only to law enforcement officials with either a valid subpoena or another valid legal authorization. Learn how to set up Call Trace and protect yourself from caller harassment.
Here's How to Use Call Trace:
If you receive a harassing call, hang up. Wait 10 seconds and then dial *57.
If the call was life threatening, or there is some other emergency, contact law enforcement immediately by dialing 911.
If the call did not rise to the level of an emergency, contact local law enforcement on the local non-emergency telephone number to file a report. Or, chat with us for further assistance.
Then call the Comcast Legal Response Center at 1-866-947-8572.
The Comcast Legal Response Center will require the following information from you:
Police Department case/report number
Officer’s name
Customer’s Name
Caller’s Name
Telephone number that is being called and number calling
The Comcast Legal Response Center will complete a Call Trace report and fax the report to the local police department to add to the case/report file.