Magnetic Bloch analysis and Bochner Laplacians (Q1326566)

The authors consider a particle in a magnetic field where the configuration space \(M\) is assumed to be a 2-torus and the magnetic field is assumed to satisfy an integrality condition. Then quantization can be established by choosing a Hermitian line bundle over \(M\) with connection \(\nabla\) such that the curvature of \(\nabla\) equals the magnetic field. The Hilbert space of quantum mechanical states is given as the space of \(L^ 2\) sections in this bundle and the Hamiltonian is given as the Bochner Laplacian \(\nabla^* \nabla\). As the 2-torus is not simply connected, the construction is non-unique. The authors show that the family of Bochner Laplacians arising in this construction is related to the so-called Bloch decomposition of the corresponding Hamiltonian on the covering space \(\mathbb{R}^ 2\) of \(M\). Applications are not worked out, but it is mentioned that motivation for this work comes from the quantum Hall effect.