Coincidence of length spectra does not imply isospectrality

Details Coincidence of length spectra does not imply isospectrality Coincidence of length spectra does not imply isospectrality

2005-05-11

arxiv.org/abs/math/0505200v1

Inverse Problems 21 (2005) 1391-1395

Mathematics

Spectral Theory

8 pages, 5 figures

Scientific paper

10.1088/0266-5611/21/4/013

Penrose--Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing ``no''). Here we show that generically (in the class of smooth domains) the two members of a mushroom pair have different spectra.

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