Title: One cannot hear the density of a drum (and further aspects of isospectrality) Show Chinese title

Title: 无法听出鼓的密度（以及等谱性的其他方面）

Abstract: It is well known that certain pairs of planar domains have the same spectra of the Laplacian operator. We prove that these domains are still isospectral for a wider class of physical problems, including the cases of heterogeneous drums and of quantum billiards in an external field. In particular we show that the isospectrality is preserved when the density or the potential are symmetric under reflections along the folding lines of the domain. These results are also confirmed numerically using the finite difference method: we find that the pairs of numerical matrices obtained in the discretization are exactly isospectral up to machine precision. Show Chinese abstract

Abstract: 众所周知，某些平面区域对的拉普拉斯算子具有相同的谱。 我们证明这些区域在更广泛的物理问题中仍然保持等谱，包括非均匀鼓和外部场中的量子弹道问题。 特别是我们展示了当密度或势能关于区域的折叠线对称时，等谱性仍然保持。 这些结果也通过有限差分法进行了数值验证：我们发现离散化得到的数值矩阵在机器精度范围内完全等谱。