I think post the main symptoms in the emergency section for proper disgnosis.
You say keeping betta in one gallon tank, was it cycled before buying fish, does it have a heater and filter? The reason say this is bettas are tropical and need to be kept at a good temperature of 77-80F. A filter is not actually necessary but it will keep nasties down in the water. The reason asking on the filter is, one gallons is very small to keep a betta in it does not allow much leeway and problems present happen quickly. If not doing very regular water changes (especially with no filter) ammonia, waste from the fish will build up in the tank, make him unwell and lower his immune system so he keeps getting ill.
I don't know much on medications myself at all as only recently treating my first ill betta. Fin rot is generally blackened tips though sometimes can be white so be careful here.
I am going to paste some bits now from a site called ##### that sums up some things your betta may have, it may be wrong but it is a starting point:
Fin Rot
Causes - Poor water conditions, Damage leading to secondary infection
Symptoms - Fins appear ragged and uneven. They may appear to be dark at the edges, but can be white. Maybe less active and not eating. May appear pale.
Treatment - 50-75% water change. Use a anti bacteria treatment such as Myxazin or Tetracycline. Again Aquarium salt may assist at the ratio stated above. Keeping the water conditions in perfect order will assist the recover and re-growth of the fins.
Advanced Fin Rot/Body Rot
The Fin rot advances quickly to destroy the fins completely. This is particularly hard to stop and attacks the body once the fins have gone.
Symptoms - Fins and or tail start rotting away, usually starting from the edge, but sometimes it starts at the base of the fin (especially dorsal) and attacks the body directly. Diseases progresses rapidly and as the tissues are being eaten away, you might see the fins bones stick out. Once fins have been consumed, rot will proceed onto the body. At this stage the disease is hard to reverse although the betta might continue to live for months if treated properly. If not treated, it will die promptly.
Treatment - Do a full water change. Preferably provide a new tank (although that’s expensive disinfectant should clear all problems just make sure you rinse it fully as disinfectant WILL kill your fish). You must combine several medications to have a chance to stop this thing. Use Ampicillin and double the dose if needed, and use at the same time tetracycline. These medications usually come in capsules. A full capsule usually treats 10 gal of water. So for a 1/2 gallon of water, open the capsule and take the right proportion of powder and sprinkle on jar water. You may steer gently with a disposable plastic spoon. Tetracycline might turn the colour of the water to a dark yellow or red. Change water every third day and add a new dose of same medication. Continue until fins/tail stop receding and start showing some new growth. It may take up to 4 weeks to work, so don’t give up. Once healing starts you should treat for one more week to be sure.
Bacterial Infection
Causes - Poor water conditions. Bacteria are always present in the water from the tap, and can get in the water from your hands and the air. It is the natural Immune system of your fish that protect it from these bacteria. Therefore, anything that lowers its immune system can leave it vunerable to bacterial infection.
Symptoms - Betta may have clamped fins, lay at bottom or at surface, not eat, lose its color, turn gray, barely swim around. In more advanced cases, its body may start developing red patches, open sores and all kinds of nasty looking stuff. Different bacteria affect fish differently.
Treatment - Do a full water change for jars or a 70% water change for tanks. (To get rid of some of the bacteria present). The tank will need a good gravel vac and the filter will need cleaning. The sick fish can be quarentined but the whole tank should be treated with a good medication. The type of medication will be determined by the symptoms.
Fungal Infection
Causes - Poor water quality, injury, Introduced (new stock)
Symptoms - White cotton wool type patches. Maybe anywhere on the fish. Maybe less active & pale. Fins stuck together and lack of appetite. The spores of the fungus can be released and infect other fish.
Treatment - 50-75% water change. Use Anti Fungal meds such as Myxazin. Aquarium salt can assist at a ratio of 1 Teaspoon per 2 1/2 UK gallons of water. Bettas normally make a full recovery.
Im not sure what to disgnose but these are probably likely suspects either being fungal or bacterial. Have you a water test kit? Can you get us some readings on the water quality and a picture may be helpful under emergencies or here to diagnose. Sorry for all the questions, it is best to find a possible source for the problem as this may need to be sorted too as well as treating the fish.