Our neighbors stole our cat! What do we do?

Question
A few months ago, when we had just moved into our home, we noticed a cat living under our porch.  Every so often and wanted to play, so we played with it on and off and bonded with it. It was dirty and underweight, and had small superficial injuries from fighting with other neighborhood cats. We noticed that it was getting tired after playing for about a minuted or so, and was extremely skinny, so we naturally started feeding it.
Over the next month or so, the cat would come to the back porch every day and eat the food we set out for it and drink water from a dish, generally hang out, play with us, etc. Its coat noticeably improved and it became much more playful as time went on, and was able to play for much longer. We took it to the vet and got it a round of vaccinations, weighed it, and determined that it was a few years old, and had already been neutered. (hooray not having to deal with more cats!), and was indeed more than a pound underweight.
When colder weather came along, we invited it inside, where it has been spending most of its time as of late. We took it to the vet again (about a month after the last appointment), it had gained a pound (up to 8 pounds), and the vet tech remarked that it looked to be in much better condition than it previously was. They gave it a round of booster shots for the vaccinations it had received earlier, and at that time we registered it and had a microchip implanted.
Over the past several days, we experimented and allowed it to stay inside continuously, even while we go shopping and do our various other tasks throughout the day (I go to work, the girlfriend mostly works from home and runs a couple errands here or there).
This afternoon, someone knocked on the door while I was at work and my girlfriend answered it. It was one of the neighbors, with a pit bull! They came inside, said that it was their cat, scooped it up, and abruptly left. Neither of us have seen it or the neighbors since.
The neighbors had mentioned the cat in passing, they called it a name and said that it hated to be inside, that their 5 dogs chased it, etc, so I didn't think that they thought of it as their cat. It didn't seem like anyone was taking care of it or feeding it before, so it just didn't seem to me like they laid much claim to it.Also, when we took it to the vet, the vet identified that the cat has a dental issue that needs to be addressed with an expensive surgical procedure. This dental issue is the result of neglect; it is stage 4 of 5 tooth reabsorbption.
What kind of recourse do I have? I have vet documentation including rabies and microchip tags, and the fact that the cats health has improved should hopefully be on my side. How screwed am I? Thank you for your input in advance.

Answer
Alex,

I expect you have no recourse! Unless you have some sort of papers to "prove" ownership, under the law, for most jurisdictions, you do not own the cat.  I expect the neighbors brought their pit bull to intimidate whoever answered the door.

Your only possible recourse may be to talk to the local animal control people to see about charging the people with neglect of the cat. On the other hand, I would expect that people of this sort could make your life very miserable if you pursue it.

If the cat does run away and the cat is take to a shelter where they check the microchip, I expect the cat would be returned to you.

If your neighbor invades your house without permission, that is breaking and entering and is a felony in most jurisdictions.

Good luck and best regards... Norm.