Schrödinger evolution of self-gravitating discs

Abstract

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

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