On the possible application of group representations to some boundary value and eigenvalue problems of certain domains without classical symmetry

Summary

It is well known that the symmetry groupG of a plane domainD may be used to derive symmetry properties of the eigenfunctions ofD, assuming the differential operator commuting with the transformations of the group. This consideration applies also to boundary value problems. They can be decomposed into partial problems corresponding to different symmetry classes (types).

Our purpose here is to consider some classes of domains without classical symmetry, but their admissible functions have ‘generalized’ symmetry properties (see [1] and [2]).

• (1) A domainD _N (Fig. 3) is a union ofN=m+n congruent subdomains (cells)Q ⁺₁ ,Q ⁺₂ , ...,Q ⁺_m ,Q ^-₁ ,Q ^-₂ , ..., andQ ^-_n .

• (2) A domainS _N (Fig. 4) is a part of a periodic domainR with the period Ω along thex direction, situated between the linesx=x ₀ andx=x ₀+NΩ. It consists ofN congruent cells. Each cell is symmetric with respect to its middle axis.

• (3) A domainS _N+(1/2) (Fig.5) consists ofN+(1/2) congruent cells.

In the present Part A we are concerned with such domainsD _N, which we call ‘domains of the first type’. In Part B we shall consider domainsS _N (‘second type’) andS _N+(1/2) (‘third type’). Finally Part C will give some numerical examples (Part B and Part C will appear also in ZAMP.)

For the domainD _N, we obtain a group of linear operators transforming the ‘admissible’ functions into one another. It is of order two.

For the domainS _N, we obtain a group of transformations acting on the functions, ‘admissible’ inR. It is cyclic and of order 2N. For the domainS _N+(1/2) this group is of order 2N+1.

The eigenvalue as well as boundary value problems can again be decomposed into different symmetry types. This is done by constructing the idempotents (projection operators) with the help of characters of group representations. Every real idempotent projects the functions, admissible inS _N (respectively inS _N+(1/2)), into such functions which belong to a corresponding symmetry class. They offer us a mean to explain those ‘generalized” symmetry properties.

These considerations apply also to domainsS _N andS _N+(1/2) where the period Ω is an arbitrary angle. If Ω is not a submultiple of π the periodic domainR lies on a covering surface, ramified at a fixed point.

Further we obtain the following result. ‘admissible’ functions, belonging to different symmetry classes, are ‘pointwise orthogonal’, i.e., the sum of their products on a certain finite set of points vanishes.

The same considerations apply to the resolution of finite difference equations (see [3]). The given problems split into several partial problems, each one corresponding to a particular (non-classical) symmetry class. In this way the number of unknowns in each partial problem is much smaller than in the initial problem.

The idea of this work came after Prof. Dr. J. Hersch had suggested to me that there might be an algebraic explanation for the ‘generalized’ symmetry property of eigenfunctions (of a certain class of domains without classical symmetry) which he had derived analytically in [1] and [2].

Herewith I would like to thank Prof. Dr. J. Hersch very much for his useful advice and suggestions. Besides I express my thankfulness to Prof. Dr. H. Brauchli, Prof. Dr. M. M. Schiffer, Prof. Dr. E. Stiefel and Prof. Dr. B. L. van der Waerden for very helpful discussions. Last, not least, I would like to thank both the Institute of Mechanics and the Institute of Machine Tools and Production of the Swiss Federal Institute of Technology (E.T.H.), Zuerich, which encouraged and supported the redaction of the present paper.

The main results have been briefly announced in [4].