On the structure of normal spiral galaxies with finite disk thickness part III: The asymptotic solution and the quantum condition of the general eigenvalue problem

Abstract

The asymptotic solution and the quantum condition of an eigen-value problem which appears in the density wave theory of the spiral galaxy are re-examined from a mathematical point of view. The dependence of the complex eigen-frequency ω on a large parameter λ is carefully discussed. It turns out that, in the expansion ω=∑^χ _i=o ωj/λj, the first term ωo must be real in order that the radiation condition may be satisfied. A related conclusion is that the quantum condition may be expressed in terms of an integral along the real axis of the complex plane of the independent variable. This form for the complex quantum condition is convenient not only for determining the complex eigen-frequency numerically, but also for drawing some physically interesting conclusions directly.