Suppose you are playing in a limit game where you have the option of betting more than the limit but your opponent only has to call the amount of the limit. Is it ever the correct strategy against an expert opponent?

Can he only win the amount he calls?

As usual, when such questions are asked, they beget other questions. To differentiate between between a normal bet and the extra sized bet, lets call the extra sized bet an xbet.

1. Can an xbet be of any size as in NL?
2. Can you call amount greater than bet, but less than xbet?
3. Can Hero decline Villains xbet and raise bet?
4. Can Hero accept Villains xbet and raise bet?
5. Can Hero decline Villains xbet and raise xbet of a lessor, equal or greater size?
6. Can Hero accept Villains xbet and raise xbet of a lessor, equal or greater size?
7. Playing 3-way, can one person call bet and the other call xbet?
8. How do questions 2 through 6 work 3-way as well?
9. Any other weird quirks we need to consider?

The extra money is just a pot sweetener.Everything else is normal

No. He wins the whole bet when he calls he standard amount.

If it's HU then the "expert" can decide that every time you put in an extra bet to just completely ignore it, and he will be better off than if you never did. So Heads-Up it can't be part of a GTO strategy. 3+handed isn't so automatically/mathematically clear, but it seems hard to conceive that it could ever be part of good balanced strategy.

Does he have to double your bet if he wants to raise?

Pokerfarian hit the nail on the head:
This is exploitive of the pot sweetener bets. It may not be maximally exploitive, but it can do no worse than break-even and sometimes do better.

My alternative interpretations were more interesting than this.

At first glance it seems that the extra money would make an opponent (expert or otherwise) more likely to call because he is getting better pot odds.
So then it would only be a good idea to offer your opponent better pot odds than he's getting anyway if you have a lock on the hand, i.e. your opponent is drawing dead.

If your opponent is not drawing dead and still has an equity share in the pot it seems stupid to add money to the pot for no return.

I don't see how offering your opponent better pot odds than he's getting anyway could be good for you, unless he is drawing dead, in which case it doesn't really matter anyway. This assumes that because we're playing limit our normal sized bet is getting called 100% of the time.

While this will never work in a HU pot for the simplistic reason described already that the opponent can choose to ignore it, I'm almost certain this can work in an multiway pot.

Basically you would need villain with a made hand, hero with a draw, and at least 1 "other player" with a draw that don't interfere with your draw. When the "other players" just barely don't have odds to call, you can sweeten the pot that then gives them odds, which in turn barely increases the "other player"'s EV, increases your EV, and decreases villain's EV.

For an example, I was thinking about something like an 8-way pot with all player's hands being face up, limit hold'em (also assume everyone is a perfect logician and has perfect river play). Villain is last to act and hero is second last. I don't care to do the math, but imagine it is a situation such that if hero bets, villain calls, everyone must barely fold due to improper pot odds. However, instead you can bet + add a sweetener, villain calls, and everyone calls with hands that don't hurt your hand but hurt villains (someone must lose due to the zero sum nature of poker). Anyway, example situation as follows:

Board reads Td9d8h2h on the turn. First guy checks with Jc4c, second guy checks with Ks9s, third guy checks with Qc5c, fourth guy checks with 7s7c, fifth guy checks with As2s, sixth guy checks 4h3h, you bet with 4d3d, villain calls with Tc3c. Everyone must fold if you bet a normal amount... however if you add a sweetener they all call. Therefore you lose the amount of the sweetener some amount of the time, but you gain 6 guys calling with hands that don't interfere with yours- clearly +EV if the sweetener is small enough relative to the bet.

No

wat? Jc4c and Qc5c are going to call with just an overcard just because they're getting slightly better odds? that makes no sense.

Mathematically, you would only do this if you had the nuts and were trying to give some kind of dead draw the proper odds to call (in their mind, obv). Theoretically, an expert opponent would know this and you could use this odd rule to bluff more effectively. So perhaps against an expert opponent you could gain enough fold equity with your bluffs to make the play +EV?

Edit: Read Pokerfarian's post and can't argue with that logic either. So I guess the answer hinges on how the expert opponent would interpret/respond to the larger bet.

It wouldnt make sense, since the opponent will adjust. So sure, he might call you down sometimes when he's drawing dead because of the great pot odds, but he will also catch your big bluffs and make a big profit from them. And that profit will be larger than the amount he will lose when he's drawing dead, since he only needs to call the standard amount.

It's worth considering that one seldom has one's opponent drawing dead - at least seldom does one know that one's opponent is drawing dead, right? Basically you need to have a boat when your opponent is on a flush or straight draw (or bottom two pair or a low set which is drawing to a lesser boat) in order for the bet to make sense.

Seems like one would seldom want to bluff in this way except against fairly inobservant opponents. Yet bluffing into inobservant opponents is not a good idea, right?