What mechanism does an isolated white dwarf use to generate energy?
A white dwarf does not generate energy, it radiates the energy it already has into space.
A white dwarf is the stellar remnant of a low mass star. After helium fusion ends, the star contracts due to gravity, until it reaches the point that only electron degeneracy can support the star.
The temperature of a degenerate white dwarf is lower than the temperature required to fuse carbon atoms. Furthermore, the star cannot be compressed to increase the temperature, so it basically becomes a static lump of mostly carbon atoms.
Slowly over time the white dwarf will radiate its left over thermal energy into space. As it does so it will cool, since there is no mechanism to replace the thermal energy. When it has cooled enough that it is no longer visible, it will become a black dwarf.
The time frame required for a white dwarf to cool this far is longer than the theoretical age of the universe, so astronomers do not expect to find any black dwarfs anytime soon.