Kac's question, planar isospectral pairs and involutions in projective space

Abstract

In a paper published in Am. Math. Mon. (1966 73 1–23), Kac asked his famous question 'Can one hear the shape of a drum?'. Gordon et al answered this question negatively by constructing planar isospectral pairs in their paper published in Invent. Math. (1992 110 1–22). Only a finite number of pairs have been constructed till now. Further in J. Phys. A: Math. Gen. (2005 38 L477–83), Giraud showed that most of the known examples can be generated from solutions of a certain equation which involves certain involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions when n = 2. In this letter we handle all dimensions, and show that no other examples arise.