LECTURE 2. THREE SYMMETRIC SPACES 369b ehavior of level curves of eigenfunctions. This question is a mathematical analogue
of physicists' questions. The nodal lines (cpn = 0) can b e seen by putting dust on
a vibrating drum. One question quantum physicists ask is: Do the nodal lines of
eigenfunctions 'Pn "scar" (accumulate) on geodesics as the eigenvalu e approaches
infinity? The answer in the arithmetical quantum chaotic situation appears to be
"No" thanks to work of Sarnak et al [ 67 ]. You can see t his in the pictures of eigen-
functions of the Laplacian for Maass wave forms. Figure 9 is a representative picture
from an older version ofD. Hejhal's website (http://www.math.umn.edu;-hejhal/ ).
Note that the scarring in Figure 6 is along horocycles.
Figure 9. Arithmetical quantum chaos for the modular group -the topog-
raphy of Maass wave forms for the modular group from Dennis Hejhal
http://www.math.umn.edu;-hejhalFigures 10 and 11 give the finite Euclidean and non-Euclidean analogu es. An-
imations of the contour maps of Figures 10 and 11 , asp grows, are to be found at
my website
http:/ /math. ucsd .edu;-a terr as / euclid .gif
and http://math.ucsd.edu;-aterras/ chaos.gif.
In Figure 10 , the level curves are finite analogues of circles. They look like
Fresnel diffraction patterns. See Goetgheluck for a discussion. Of course, there is
no real reason to stick to our favorite finite analogue of Euclidean distance. Perla
Myers [ 62 ] has considered finite Euclidean graphs with more general distan ces, for
which the "level curves" also give quite beautiful figures. See also Bannai et al [7].
Figure 11 should be compared with Figure 9.