Thank you for your question and I'm very sorry to hear about your situation.
Your neighbor is guilty of the tort of defamation. Defamation is a written or verbal untrue statement to a third party about another which tends to or actually does cause economic damage or loss to reputation. As her actions trigger this claim, you could file the claim in court and seek either damages (in the form of any economic loss that you've experienced as a result of her statements, or as a punitive damage to her) and injunctive relief.
While you have this claim, it really is not worth pursuing unless you've experienced an actual loss or your reputation is actually damaged. In this case, it does not seem that either is the case. You've stated that no one believes her, and you've not mentioned any financial loss that her statements have caused. Accordingly, I would not recommend going through the process of filing a legal claim at this point. Further, I would not recommend sending any sort of cease or desist letter, as this is only likely to inflame her and make her state that you are continuing to threaten her.
I say this with reservation, as it really is a judgment call on your part. If you feel that her continued defamation is a serious problem, then you should first send a Cease and Desist letter demanding tht she immediately stop all defamatory actions, and that if she does not then you will initiate legal proceedings.
As far as whether you should be worried or not, I would advise you to be very careful around her. The last thing you want is for her to report you to the police. Try to avoid her and/or document every interaction with her (by video tape or audio recording if possible). This way if she ever makes any kind of criminal complaint about you, you can have actual evidence that you did not threaten her.
Please let me know if you have further questions.