Is this an internet dating scam?

My step-kids' mother is convinced she has met the man of her dreams online. He supposedly is a neurologist living in Britain and is already telling her to find a house for them with a price range up to $600,000. Everything that has happened so far has been screaming scam. Then, she did the worst thing she can do and gave him her checking account number (which she claims she did on her own to "test" him..don't ask..she's not all there). She didn't have anything in her account and then he just up and deposits $1000 in her account. That is what has me thrown. The only thing I can think of is that it is just a tactic to gain her trust and he plans to get something out of her later. But now she is convinced more than ever that he is legit and is going to come to the US and rescue her from her bad situation. But I just can not imagine someone with as much money as he supposedly has looking online for a date and then actually dropping everything to move to another country for her. Her kids are actually starting to believe it and I don't want to see them let down. But I look like the mean party pooper for trying to warn them. So, does anyone have any insight as to why someone would deposit that kind of money if it is a scam? It wasn't a money order...it was a direct deposit.