Does general anesthesia have a long-term effect on the memory after the operation?

Yes there is some: Evidence that this can happen. More so depending on the patient, and more true as you get older and have more chronic medical problems.

Possibly: There is some research to suggest that this is the case. If so, it is probably more prominent in longer operations, multiple operations, and worse the older you get. The "flurane" inhaled anesthetics seem to be the main culprits. It is unknown if there's any way to prevent this. Which might be hard, since we don't even know how those drugs work yet!

No: Perhaps if during the induction stage suggestions are made, much like a hypnotic stage, long term memory may be affected.

Rarely: There is evidence to suggest that rarely general anesthesia can result in long-term memory loss. This is mostly found in very elderly patients having major operations such as heart surgery and joint replacement. Millions of patients have had general anesthesia with no problems whatsoever in the postoperative period. Younger patients typically wake up feeling exactly the same as before.

Unlikely: Memory issues are seen rarely and then mainly in the elderly.

Rarely: The anesthesia can have adverse effects on the central nervous system in the extremes of age. It does not usually have problems in most patients. The question is wether it is the anesthetic agents, or wether it is that patients have so little reserve that a drop in blood pressure can cause problems. Anesthesiologists specializing in geriatric patients can help minimize these problems.