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Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians. (arXiv:1810.05623v1 [math-ph])
来源于:arXiv
It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a
constant magnetic field $b$ which generates a magnetic flux $\varphi$ per unit
area, then any spectral island $\sigma_b$ consisting of $M$ infinitely
degenerate Landau levels carries an integrated density of states
$\mathcal{I}_b=M \varphi$. Wannier later discovered a similar Diophantine
relation expressing the integrated density of states of a gapped group of bands
of the Hofstadter Hamiltonian as a linear function of the magnetic field flux
with integer slope.
We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau
operator $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electric
potential. Assume that $H_b$ has a spectral island $\sigma_b$ which remains
isolated from the rest of the spectrum as long as $\varphi$ lies in a compact
interval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ on such
intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$ 查看全文>>