How much do Springer-Verlag authors make per book sold?
This varies substantially, depending on the sort of book, how well the author negotiated (each book involves an individually negotiated contract), the price of the book, etc. Based on one Springer contract I'm familiar with, here's a first approximation. I can't say for sure how representative it is, but it's at least one data point, and I'd guess it's typical for Springer and competing publishers.
For regular sales, the author may get 9-12% of "net cash receipts", defined as what the publisher made on the sale (excluding taxes), with the percentage depending on things like how well the book has sold. Note that "regular sales" excludes some special cases, for which there may be lower royalty rates.
Then the question is how much Springer charged the bookseller for the book. You can download a 6 megabyte zip file of Springer wholesale prices. I glanced at a few math books, and the prices look like they are about 60% of what bookstores are charging. That's certainly the right ballpark, but I haven't computed any real statistics.
So a first approximation is that the author's royalty is probably 5-7% of the price you pay in a bookstore. Considering that academic books are expensive, that's not bad, but nobody's going to grow rich off it. For advanced books, selling 1000 copies in total is very good and selling 10,000 is amazing.