Geometrical interpretation of a Schrödinger operator 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 $m_{-}, m_{+} > 0$. 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 $L^{2}(\mathbb{R}^{2})$ and essentially selfadjoint in $C_{0}^{\infty}(\mathbb{R}^{2})$.
If $g(x) = I$ for all $x \in \mathbb{R}^{2}$, this operator is called the Landau Hamiltonian and has the quantum mechanical interpretation of a spinless, charged particle under a constant magnetic field $b > 0$. I would like to 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?
 A: I'm afraid I don't know any references off the top of my head, but let me at least give you a differential-geometric translation of sorts.


*

*If $g$ is real-valued, then it defines a Riemannian metric on $\mathbb{R}^2$.

*The vector field $A$ is a vector potential for the constant magnetic field $b$, in the sense that $\operatorname{curl}(A) \cdot \hat{\mathbf{k}} = b$. Even better, the magnetic potential $A$ can be interpreted as the connection $1$-form
$$
 A := -\frac{1}{2}bx^2\mathrm{d}x^1 + \frac{1}{2}bx^1\mathrm{d}x^2
$$
of the unitary connection $\nabla = d - iA$ on the (trivial) line bundle $\mathbb{R}^2 \times \mathbb{C}$ defined by
$$
 \nabla_{\frac{\partial}{\partial x^1}} = \frac{\partial}{\partial x^1} + i \frac{1}{2}bx^2, \quad \nabla_{\frac{\partial}{\partial x^2}} = \frac{\partial}{\partial x^2} - i \frac{1}{2}bx^1,
$$
with curvature (viz, magnetic field strength)
$$
 B := \mathrm{d}A + A \wedge A = \mathrm{d}A = b \,\mathrm{d}x^1 \wedge \mathrm{d}x^2;
$$
observe that sections of the line bundle $\mathbb{R}^2 \times \mathbb{C}$ are just complex-valued functions on $\mathbb{R}^2$. Parallel transport along this connection, then, is none other than magnetic translation.

*Given a complete Riemannian manifold $(X,g)$, a Hermitian vector bundle $E \to X$, and a unitary connection $\nabla$ on $E$, you get a canonical essentially self-adjoint second order elliptic differential operator called the Bochner Laplacian $\nabla^\ast \nabla$, which satisfies, as one would expect,
$$
 \nabla^\ast \nabla = \sum_{j,k} g^{jk}(x)\nabla_{\frac{\partial}{\partial x^j}}\nabla_{\frac{\partial}{\partial x^k}} + \text{lower order terms}.
$$
If I'm not too mistaken, your magnetic Schroedinger operator $H$ should basically be the Bochner Laplacian for the connection $\nabla = d - iA$ on the trivial line bundle $\mathbb{R}^2 \times \mathbb{C}$ over $(\mathbb{R}^2,g)$.


I might have some signs and constants out of place here or there, but I think this should give at least the moral translation into differential-geometric terms. Lurking in the background is Weyl's famous characterisation of electromagnetism as a  $U(1)$ gauge theory, i.e., in terms of a unitary connection (viz, electromagnetic potential) on a line bundle over spacetime, whose curvature therefore gives the electromagnetic field strength.
A: your $\mathbf{g}$ is the position-dependent inverse effective mass tensor; if you calculate the classical equations of motion
$$
{{\ddot{x}_1}\choose{\ddot{x}_2}}=2b\,\mathbf{g}(\mathbf{x})\cdot{{-\dot{x}_2}\choose{\dot{x}_1}}
$$
then the semiclassical quantization condition is that the area inside a closed orbit must equal $(2\pi/b)(n-1/2)$, $n=1,2,3,\ldots$.
For a discussion in a three-dimensional setting, but without the position dependence of $\mathbf{g}$, see these lecture notes.
For a physical application to a semiconductor with position dependent effective mass, see Landau levels and cyclotron resonance in graded mixed semiconductors.
