It sounds like it's time to consult a real estate attorney. Property disputes like this can be very difficult and emotion-fueled and people are often not reasonable when it comes to their own property (remember that saying about a man's home being his castle), even if they are in the wrong. Hopefully, a good real estate attorney can negotiate a settlement without dragging this into court, but you'll need to know about the liabilities, potential claims, and damages involved. If negotiation doesn't work, then the next step is probably filing a civil suit for injunctive relief and money damages.
My responses to posts on AVVO are not legal advice, nor do they create an attorney-client relationship. In order to provide true (and reliable) legal advice, an attorney must be able to ask questions of the person seeking legal advice and to thus gather the appropriate information. In order for an attorney-client relationship to exist, you and I both have to agree the the terms of such an agreement.
I am not an OR attorney, laws vary from state to state, you should therefore consult a local attorney.
As a general rule, you are allowed to cut any branches or roots that are on your propert.
Gostina v Ryland, 199 P. 298 (Wash. 1921): "One adjoining owner cannot maintain an action against another for the intrusion of roots or branches of a tree which is not poisonous or noxious in its nature. His remedy in such cases is to clip or lop off the branches or cut the roots at the line."
If this answer was helpful, please mark it as helpful or as a best answer. This answer is for general education purposes only. It neither creates an attorney-client relationship nor provides legal guidance or advice. The answer is based on the limited information provided and the answer might be different had additional information been provided. You should consult an attorney.
I believer there are court decisions supporting the right of an adjacent landowner to cut the branches of a tree which intrudes onto his property. I suspect the same rule applies to roots, but I am not aware of any decisions on roots.
I think the practical problem is trying to get at the roots to sever them, and the fact that removing them may kill the tree. I suggest you contact a property lawyer in your area. The lawyer might inform the neighbor of your intent to sever the roots, and a contact from a lawyer may convince the neighbor that you are serious, and get some response.
This comment is general in nature and is not intended as legal advice. It does not create an attorney client relationship and obviously is not confidential. You should contact an attorney in your area who can review with you all of the relevant facts and give you specific legal advice.