Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some $m_{-}, m_{+} > 0$, where $\mathbb{C}^{2}$ is endowed with the usual inner product. Set $A = (A_{1}, A_{2}) := (-bx_{2}/2, bx_{1}/2)$, $b > 0$ and define the operator $$H = \sum_{j,k=1,2} (-i\partial/\partial x_{j} - A_{j}) g_{jk} (-i\partial/\partial x_{k} - A_{k}),$$ selfadjoint in $L^{2}(\mathbb{R}^{2})$.

If $g(x) = I$ for all $x \in \mathbb{R}^{2}$, this operator is simply the Landau Hamiltonian and has the quantum mechanical interpretation of a spinless, charged particle under a constant magnetic field $b > 0$. I would be very glad if you could help me understand further physical interpretations for the operator $H$ (in its general form).

Also, I understand that this operator can be given an interpretation in geometrical terms. Could you please explicit these interpretations and give some references to better understand them?



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