  An understanding of the long-term evolution of self-gravitating disks ranks
among the classic outstanding problems of astrophysics. In this work, we show
that the secular inclination dynamics of a geometrically thin quasi-Keplerian
disk, with a surface density profile that scales as the inverse square-root of
the orbital radius, are described by the time-dependent Schr\"odinger equation.
Within the context of this formalism, nodal bending waves correspond to the
eigenmodes of a quasiparticle's wavefunction, confined in an infinite square
well with boundaries given by the radial extent of the disk. We further show
that external secular perturbations upon self-gravitating disks exhibit a
mathematical similarity to quantum scattering theory. Employing this framework,
we derive an analytic criterion for the gravitational rigidity of a
nearly-Keplerian disk under external perturbations. Applications of the theory
to circumstellar disks and Galactic nuclei are discussed.
