Before answering a question like this, one must know the background, so I found
your other thread.
The law will vary from state to state, and from city to city. In this case, the law does not matter though.
You should not patrol your property with a shotgun unless you intend to use it if necessary. The offense that has been committed does not meet the requirements for lethal force. If you use the shotgun, you will likely not remain a free man and your girls will not have a dad in the home. They will then live next to your neighbor without a dad.
If you do not use the shotgun, your carrying it in this instance, is to threaten and intimidate. That is called brandishing. Brandishing a firearm is illegal. Expect to loose some time with your girls. You will at the least suffer financially.
So what can be done? Well, if you were in Louisiana, your neighbor would have already committed a felony. You could prosecute and jail him for the actions he has already committed. I assume you do not live in a civilized state like Louisiana (tounge firmly in cheek).
Try a board fence.
Try an attorney.
Try moving.
Try the local dirt digging investigative reporter.
Try prosecuting him for the assault and battery on yourself with a rake.
But don't try to intimidate him with a gun, because you will not win. It's time to start learning to use the law more than a gun, and I don't mean by gathering information on the internet, from a cop or the gun counter.