What causes a breathy voice when singing, if there is no apparent problem with the vocal cords?
More than likely this is happening because the arytenoid cartilage is rotated/shifted by pressure in the front cervical spine and the glottis is easily opened due to deep positioning.
The following figure supports this theory:
The black arrow is press to the depth of space,the red arrow is press from the cervical spine and the blue arrow is opening force of vocal cord.
In reference to the above picture,
You can stop and/or prevent having a breathy voice, if you fix this deep position.
If you go to an otological hospital but they cannot find any problems with your vocal cords, and you still have a breathy voice, then I advise you to seek second opinion through a medical professional who can check the position of your larynx.
Note: Many people who have breathy voices are usually only recognized by people who have a good ear. Most commonly it is also observed when recording a voice, playback, or simply while singing using a microphone.
Please consult with professional who is educated in all aspects of the larynx, both medically and through professional singing.