Geodesic
Periodic Magnetic Schr\"odinger Operators: Spectral Gaps and Tunneling Effect
Informatics and Automation, 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{TRSPY_2008_261_a12,
     author = {Yu. A. Kordyukov and B. Helffer},
     title = {Periodic {Magnetic} {Schr\"odinger} {Operators:} {Spectral} {Gaps} and {Tunneling} {Effect}},
     journal = {Informatics and Automation},
     pages = {176--187},
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     volume = {261},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a12/}
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Yu. A. Kordyukov; B. Helffer. Periodic Magnetic Schr\"odinger Operators: Spectral Gaps and Tunneling Effect. Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 176-187. http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a12/