Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrdinger Operators

Let $Gamma=q_1mathbb{Z}oplus q_2 mathbb{Z}opluscdotsoplus q_dmathbb{Z}$, with $q_jin (mathbb{Z}^+)^d$ for each $jin {1,ldots,d}$, and denote by $Delta$ the discrete Laplacian on $ell^2left( mathbb{Z}^d
ight)$. Using Macaulay2, we first numerically find complex-valued $Gamma$-periodic potentials $V:mathbb{Z}^d o mathbb{C}$ such that the operators $Delta+V$ and $Delta$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.

Faust, M., Liu, W., Matos, R., Plute, J., Robinson, J., Tao, Y., Tran, E., & Zhuang, C.

Faust, Matthew||Liu, Wencai||Matos, Rodrigo||Plute, Jenna||Robinson, Jonah||Tao, Yichen||Tran, Ethan||Zhuang, Cindy

Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrdinger Operators Institutional Repository Document