Isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to iso-length spectrality of the corresponding domains. Although an explicit and exact trace formula such as Poisson's summation formula or Selberg's trace formula is not known to exist for such planar domains, equivalence between isospectrality and iso-length spectrality in a certain setting can be proved by employing the matrix representation of "transplantation of eigenfunctions" [1]. As an application of the equivalence, transplantable pairs of domains, which are all isospectral pair of planar domains and therefore counter examples of Kac's question "can one hear the shape of a drum?", are numerically enumerated and it is found at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to Sunada [1,2]
References
[1] P.Buser, J.Conway, P.Doyle and K.D.Semmler, Internat. Math. Res. Notices 9 (1994)
[2] T.Sunada, Ann. Math. 121 (1985)
[3] Y.Okada and A.Shudo, J. Phys. A. 34 (2001)
The quantum billiard problem, that is the Dirichlet problem for the Helmholtz equation, can be rewritten as a Fredholm integral equation of the second kind with the aid of the Green's theorem, and then the eigenenergies of the quantum billiard can be characterized as the zeros of the Fredholm determinant on the real axis [1,2]. From this view point, we pose a new question "can one determine the shape of billiard table through the Fredholm determinant?" instead of the famous Kac's question "can one determine the shape through the eigenenergies i.e. the zeros of the Fredholm determinant?" [3], which was solved negatively [4]. Our numerical tests reveal that the answer to our question addressed above is "yes", that is, the shapes of the isospectral pair of billiards are distinguishable from the eyes of the Fredholm determinants although they have exactly the same eigenenegies [5]. Via "the interior-exterior factorization of the Fredholm determinant" [2], the difference of the Fredholm determinants between the isospectral pair can be interpreted as the difference of the scattering phase shifts when the billiard table are regarded as a scatterer against the exterior wave function.
References
[1] B.Georgeot and R.E.Prange, Phys. Rev. Lett. 74 (1995)
[2] S.Tasaki, T.Harayama and A.Shudo, Phys. Rev. E 56 (1997)
[3] M.Kac, Am. Math. Monthly 73 (1966)
[4] C.Gordon, D.Webb and S.Wolpert, Invent. math. 110 (1992)
[5] Y.Okada, A.Shudo, S.Tasaki and T.Harayama, to be submitted