An introduction to 3D discrete magnetic Laplacians and noncommutative 3-tori
Let \(\Theta= (\theta_1, \theta_2, \theta_3)\), \(\beta= [0,1]\), and \(c= (c_1,c_2,c_3)\). The 3D discrete magnetic Laplacian, or 3D-DML, is defined as \[ \Delta_{\theta,c}= c_1(U^*_1+ U_1)+ c_2(U^*_2+ U_2)+ c_3(U^*_3+ U_3), \] given by \(U_1\xi(x)= \exp\{i\pi\Theta\cdot (-x\times e_j)\}\xi(x- e_j)\) for \(\xi(x)\in \ell^2(\mathbb{Z}^3)\). Let \(S(\Theta,c)\) be the spectrum \(\text{Sp}(\Delta_{\theta,c})\). The author studies here the noncommutative 3-torus (i.e. the \(C^*\)-subalgebra \(A_\theta\) of \(B(\ell^2(\mathbb{Z}^3))\) generated by \(U_j\); \(j= 1,2,3\)), and the properties of \(S(\Theta,c)\). Next, the Mathieu operator \(h_{\theta,c,\beta_2,\beta_3}\) is defined as follows: \(h_{\theta,c,\beta_2,\beta_3}= c_1\Delta+ c_2 M_{\theta_3,\beta_3}+ c_3 M_{\theta_2,\beta_2}\), where \((\Delta\psi)(m)= \psi(m+ 1)+\psi(m- 1)\), and \(M_{\theta,\beta}\) the multiplication operator associated to \(m\to 2\cos(2\pi(m\theta+ \beta))\). \(\Theta\) is said to be degenerate, if \(1\), \(\theta_2\) and \(\theta_3\) are rationally dependent. He obtains the following analogous result to the one in the 2D case: \(S(\theta,c)= \text{Sp}(h_{\theta, c,\beta_2,\beta_3})\), if \(\Theta\) is nondegenerate. The spectrum of a 3D-DML could always be a Cantor set in the nondegenerate case.
scientific article; zbMATH DE number 1326886
scientific article