  We investigate the nature of low T/W dynamical instabilities in
differentially rotating stars by means of linear perturbation. Here, T and W
represent rotational kinetic energy and the gravitational binding energy of the
star. This is the first attempt to investigate low T/W dynamical instabilities
as a complete set of the eigenvalue problem. Our equilibrium configuration has
"constant" specific angular momentum distribution, which potentially contains a
singular solution in the perturbed enthalpy at corotation radius in linear
perturbation. We find the unstable normal modes of differentially rotating
stars by solving the eigenvalue problem along the equatorial plane of the star,
imposing the regularity condition on the center and the vanished enthalpy at
the oscillating equatorial surface. We find that the existing pulsation modes
become unstable due to the existence of the corotation radius inside the star.
The feature of the unstable mode eigenfrequency and its eigenfunction in the
linear analysis roughly agrees with that in three-dimensional hydrodynamical
simulations in Newtonian gravity. Therefore, our normal mode analysis in the
equatorial motion proves valid to find the unstable equilibrium stars
efficiently. Moreover, the nature of the eigenfunction that oscillates between
corotation and the surface radius for unstable stars requires reinterpretation
of the pulsation modes in differentially rotating stars.
