Geodesic
On discreteness of the spectrum of some operator sheaves associated with a periodic Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 74 (1988) no. 1, pp. 94-102 Cet article a éte moissonné depuis la source Math-Net.Ru

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Three-dimensional periodic Schrödinger operators with potentials that are square integrable on the unit cell (single-electron model of a crystal) are considered. A description is given of the class of rational curves that do not have more than a finite number of common points with any isoenergy surface (in particular, the Fermi surface) of an arbitrary operator of the considered form. A consequence of a theorem proved in the paper is the absence on the isoenergy surfaces of elements of planes, cones, and cylinders with straight generators, and all possible paraboloids and hyperboloids. Another interesting consequence is the following assertion: The topological dimension of an isoenergy manifold does not exceed two, which justifies the use of the word “surface”. The results generalize the assertion of Thomas's theorem on the absence on isoenergy surfaces of straight edges. 
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     author = {V. V. Dyakin and S. I. Petrukhnovskii},
     title = {On discreteness of~the~spectrum of~some operator sheaves associated with a~periodic {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {94--102},
     year = {1988},
     volume = {74},
     number = {1},
     language = {ru},
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}
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V. V. Dyakin; S. I. Petrukhnovskii. On discreteness of the spectrum of some operator sheaves associated with a periodic Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 74 (1988) no. 1, pp. 94-102. http://geodesic.mathdoc.fr/item/TMF_1988_74_1_a7/

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