Click here for some groovy animation in Live3D.
That is, u and v are universal isometries which commute up to a phase factor of angle theta, with A_theta the universal C*-algebra generated by u and v.
The spectrum of the operator h is a subset of the real line; the operator h changes as the angle theta varies, giving different spectral sets as well.Stacking these sets up vertically gives the two dimensional picture as in Mathieu Spectral Viewer.
Another physical quantity "lambda" can be introduced, which measures the interaction level in the electron-lattice system. By letting lambda vary as well, we get the three dimensional picture below, which represents the various spectral lines of the operator
h = u + u* + lambda (v+v*).
By the way, this picture is an interactive three dimensional image rendered in VRML; the first versions were drafted by my student S. Locke in QuickDraw 3D. If your browser supports VRML, then you can "drive around" the picture and view it from a variety of angles.
Click here for some groovy animation in Live3D.
If
you're feeling adventurous, you might try clicking here to see an interesting
picture of polygons, and figure out its relationship to the above collection
of cylinders.
The computations and resulting images above have been considerably simplified so they run on most browsers. You might try you luck with some more detailed (but complex) images below.
Click here for a 3D image with more lambda values.
Click here for a 3D image with more theta values.
Click here to a 3D image with more lambda and more theta values.
Note: many VRML viewers have problems with drawing large, complex diagrams that are generated for mathematical visualization.
For instance, Live3D has a problem with drawing single lines, preferring to instead draw double lines. To make up for this, we draw solid cylinders to represent the spectral lines, but this is clearly a poor hack for the real thing. Other viewers choke on large files with lots of small components (like spectral cylinders); thus we keep the number of spectral lines low, sacrificing accuracy for reasonable performance. Our picture with polygons is interesting, but hard to see through since they never really are very transparent.
