EXPERIMENTS ON NOT HEARING THE SHAPE OF DRUMS

We present an experimental test of the theorem of isospectral domains which states that certain pairs of planar domains have identical spect ra for eigenvalues of the Laplacian operator or the Helmholtz wave equ ation. The experiments employ thin microwave cavities shaped in the fo rm of two different domains known to be isospectral. We verify the equ ality of at least 54 of the measured low-lying eigenvalues to a few pa rts in 10(4). We obtain the eigenfunctions experimentally, and confirm that two non-isometric transformations connect isospectral eigenfunct ion pairs. An analysis in terms of periodic orbits using the trace for mula is carried out, and shown to be an alternative confirmation of is ospectrality.