Yu Pei, Dietz Barbara, Xu Hong-Ya, Ying Lei, Huang Liang, Lai Ying-Cheng
School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China.
School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA.
Phys Rev E. 2020 Mar;101(3-1):032215. doi: 10.1103/PhysRevE.101.032215.
"Can one hear the shape of a drum?" Kac raised this famous question in 1966, referring to the possibility of the existence of nonisometric planar domains with identical Dirichlet eigenvalue spectra of the Laplacian. Pairs of nonisometric isospectral billiards were eventually found by employing the transplantation method which was deduced from Sunada's theorem. Our main focus is the question to what extent isospectrality of nonrelativistic quantum billiards is present in the corresponding relativistic case, i.e., for massless spin-1/2 particles governed by the Dirac equation and confined to a domain of corresponding shape by imposing boundary conditions on the wave function components. We consider those for neutrino billiards [Berry and Mondragon, Proc. R. Soc. London A 412, 53 (1987)2053-916910.1098/rspa.1987.0080] and demonstrate that the transplantation method fails and thus isospectrality is lost when changing from the nonrelativistic to the relativistic case. To confirm this we compute the eigenvalues of pairs of neutrino billiards with the shapes of various billiards which are known to be isospectral in the nonrelativistic limit. Furthermore, we investigate their spectral properties, in particular, to find out whether not only their eigenvalues but also the fluctuations in their spectra and their length spectra differ.
“人们能听出鼓的形状吗？”卡茨在1966年提出了这个著名的问题，指的是存在具有相同拉普拉斯算子狄利克雷特征值谱的非等距平面区域的可能性。通过采用从周田定理推导出来的移植方法，最终找到了非等距等谱台球对。我们主要关注的问题是，在相应的相对论情形下，即对于由狄拉克方程描述且通过对波函数分量施加边界条件而限制在相应形状区域内的无质量自旋1/2粒子，非相对论量子台球的等谱性在多大程度上存在。我们考虑中微子台球的情况[贝里和蒙德拉贡，《英国皇家学会学报A》412, 53 (1987)2053 - 916910.1098/rspa.1987.0080]，并证明当从非相对论情形转变到相对论情形时，移植方法失效，因而等谱性丧失。为了证实这一点，我们计算了具有各种台球形状的中微子台球对的特征值，这些台球在非相对论极限下已知是等谱的。此外，我们研究它们的谱性质，特别是要弄清楚是否不仅它们的特征值，而且它们谱中的涨落以及它们的长度谱都有所不同。
