Geodesic
Periodic Magnetic Schr\"odinger Operators: Spectral Gaps and Tunneling Effect
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 176-187

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A periodic Schrödinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M,\mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field, the existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system. 
@article{TM_2008_261_a12,
     author = {Yu. A. Kordyukov and B. Helffer},
     title = {Periodic {Magnetic} {Schr\"odinger} {Operators:} {Spectral} {Gaps} and {Tunneling} {Effect}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {176--187},
     publisher = {mathdoc},
     volume = {261},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_261_a12/}
}
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Yu. A. Kordyukov; B. Helffer. Periodic Magnetic Schr\"odinger Operators: Spectral Gaps and Tunneling Effect. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 176-187. http://geodesic.mathdoc.fr/item/TM_2008_261_a12/