Geodesic
Asymptotic Behavior of the Eigenvalues of the Schr\"odinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder
Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 656-664

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In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrödinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum. 
@article{MZM_2005_77_5_a1,
     author = {V. V. Grushin},
     title = {Asymptotic {Behavior} of the {Eigenvalues} of the {Schr\"odinger} {Operator} with {Transversal} {Potential} in a {Weakly} {Curved} {Infinite} {Cylinder}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {656--664},
     publisher = {mathdoc},
     volume = {77},
     number = {5},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_5_a1/}
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V. V. Grushin. Asymptotic Behavior of the Eigenvalues of the Schr\"odinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder. Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 656-664. http://geodesic.mathdoc.fr/item/MZM_2005_77_5_a1/