The linear eigenvalue problem for barotropic selfgravitating discs
  Gaseous rotating razor-thin discs are a testing ground for theories of spiral
structure that try to explain appearance and diversity of disc galaxy patterns.
These patterns are believed to arise spontaneously under the action of
gravitational instability, but calculations of its characteristics in the gas
are mostly obscured, presumably due to a difficult outer boundary condition.
The paper suggests a new effective method for finding the spiral patterns based
on an expansion of small amplitude perturbations over finite radial elements.
The final matrix equation is extracted from the original hydrodynamical
equations without the use of an approximate theory and has a form of the linear
algebraic eigenvalue problem. The method is applied to an exactly solvable
model with finite outer boundary and to a galactic disc model.
