Two months ago I registered trial account at Springer professional digital library, downloaded few stuff and stop using their service. I was assuming that after the trial period ends the account becomes inactive.
Never assume. Always read what you're signing up for and agreeing to. You wouldn't sign a contract without knowing what you were putting your name to, this is no different. Consider this a lesson learned.
Trials like this are very common and the onus is on you to cancel a trial subscription before it carries over into a non-free account.
Cancel account and don't pay a dime. I doubt they would be able to even prove it was you who registered.
I'd be very careful with following this advice. Depending on what personal details and proof of ID you have given them you may find yourself in a bit of a hard place. I'm not sure what country you are in so you'd need to check your local laws but to give you an example, in Australia if you sign up to something, have a contract or a loan and have an account/invoice that is more than 14 days in arrears, then that missed payment will be reported to our credit reporting agencies. Leave it longer and the matter just gets worse and you start getting defaults recorded against your name. You could find it very difficult or impossible to obtain credit or even open a telephone contract in the future. All they need is your name, address and/or date of birth. Not to mention the company can sell the debt to debt collection companies who will start hassling you or even worse, obtaining court orders to begin repossessing your belongings in order to recoup the amount.
The best thing you can do at this stage is contact the company and come clean. Tell them you didn't read what you were signing up to and your understanding of the free trial was not what you expected or assumed. See if they will be nice enough to close your account and wipe the slate clean. Doing nothing and pretending it will go away usually ends up worse for you.