Astron. Astrophys. 321, 1024-1026 (1997)

3. The Hilbert space of the displacement field

Let be a function space whose elements are and in which the inner product is defined as

where the integration is over the volume of the star. on is self adjoint, . There follows the eigenvalue problem

where is the eigenfrequency of an oscillation mode, is its eigendisplacement vector, and n is a collection of three indices, indicating the three wave numbers in, say, directions of a spherical polar coordinates. Furthermore, is an orthogonal set and can be normalized to unity,

The set is also complete and may serve as a basis for . See Dixit et al. (1980). Thus, any may be expanded in a unique way in terms of . Numerical values of for hypothetical or actual star models are abundant in the astronomical literature of seventies and eighties.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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