Common problems for bettas in vases (which is why you should avoid keeping them in there):
- Unclean water, either from too few water changes or - if in with a plant - rotting root matter
- Malnutrition (owner assumes betta eats roots)
- Temperature fluctuations; vases are not typically heated, and equally as seldom have a thermometer to moniter temp. Chances are, if the betta is lethargic, he is way too cold; unless your room temp is over 70, you can safely assume he is too cold. Bettas need a temp of at least 72 to remain healthy, and even that is pushing it.
- Not enough space. Too little water means higher waste concentrations, not enough exersize less behavioral enrichment, greater fluctuation in temperature, etc.
Cold temps and little exersize a recipie for disaster. Bettas who are cold and inactive are very prone to constipation, which can be fatal if not adressed promptly. I would make a guess that the problem is one or all of the above; might we ask how much you feed, how often you clean, and what temp the vase is?