Tyler Cowen says yes and predicts that over 1,000,000 Haitians will be living in (I assume he means US run) "refugee camps for the foreseeable future". He closes with the line that "everyone in Haiti is looking to president Obama".
I know that Tyler has a deep and special relationship with Haiti. He introduced me and Mrs. Angus to the work of Antoine Oleyant and Edouard Duval-Carrie. But I personally do not foresee the US getting involved in a huge long term way in Haiti.
One reason is our history. I don't see how we can get away running things there. Could a mission there survive if American soldiers killed Haitian people?
Another reason is politics. Haitians don't vote. I realize there are a fair number of Haitians living in the USA but they are not a powerful lobby and I don't think are likely to become one. It's not clear to me what groups would favor the US taking the lead in running and reconstructing Haiti.
Another reason is that, as Tyler himself points out, it's a no-win situation. If we do take over, we are unlikely to be successful. Obama probably does not want to "preside over the collapse of a country of more than nine million people" but, contra Tyler, I don't understand why he has to or will do so.
Reconstructing Haiti needs to go through the World Bank or the UN. Not because those institutions would do a better job, but simply because the US can't / won't / shouldn't be seen as the party that bears the responsibility for the outcome.