Geometric dynamics on quantum Nano-surfaces and low-energy spectrum in a homogeneous magnetic field

Low-energy dynamics and spectrum of the Schrödinger operator on curved surfaces in a magnetic field are considered. Nano-and micro-algebraic structures generate a quantization of the surface. On this quantum space, the dynamics has a “surfing” behavior described via curvature invariants. Two types of surface deformations correspond to trapping and transistor types of dynamics.We observe the “negative mass” effect in the low-energy spectrum under trapping deformations. We also compute, in geometrical terms, the energy levels near the spectral bottom (these levels are below the minimal Landau level). The surfing dynamics and the spectrum are independent of the magnitude of the magnetic field.