Abstract
In the present paper, we obtain an asymptotic expansion of the eigenvalues of the Schrödinger operator with the magnetic field taken into account and with zero Dirichlet conditions in closed tubes, i.e., in closed curved cylinders with intrinsic torsion under uniform compression of the transverse cross-sections, with respect to a small parameter characterizing the tube’s transverse dimensions. We propose a method for reducing the eigenvalue problem to the problem of solving an implicit equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. P. Maslov, “An asymptotic expression for the eigenfunctions of the equation Δu + k 2 u = 0 with boundary conditions on equidistant curves and the scattering of electromagnetic waves in a waveguide,” Dokl. Akad. Nauk SSSR 123(3), 631–633 (1958) [Soviet Phys. Dokl. 3, 1132–1135 (1959)].
V. P. Maslov, “Mathematical aspects of integral optics,” Russ. J. Math. Phys. 8(1), 83–105 (2001).
V. P. Maslov and E. M. Vorob’ev, “On single-mode open resonators,” Dokl. Akad. Nauk SSSR 179(3), 558–561 (1968).
V. V. Belov, S. Yu. Dobrokhotov, and S. O. Sinitsyn, “Asymptotic solutions of the Schrödinger equation in thin tubes,” Trudy Inst. Matem. Mekh. Ural Otd. RAS 9(1), 1–11 (2003).
V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, and T. Y. Tudorovskii, “Semiclassical approximation and the canonical Maslov operator for nonrelativistic equations of quantum mechanics in nanotubes,” Dokl. Ross. Akad. Nauk 393(4), 460–464 (2003) [Russian Acad. Sci. Dokl. Math.].
V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskii, “Quantum and classical dynamics of electron in thin curved tubes with spin and external electromagnetic fields taken into account,” Russ. J. Math. Phys. 11(1), 109–118 (2004).
V. V. Belov, S. Yu. Dobrokhotov, and T. Y. Tudorovskii, “Asymptotic solutions of nonrelativistic equations of quantum mechanics in curved nanotubes. I. Reduction to spatially one-dimensional equations,” Teoret. Mat. Fiz. 141(2), 267–303 (2004) [Theoret. and Math. Phys. 141 (2), 1562–1592 (2004)].
V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, and T. Y. Tudorovskii, “Generalized adiabatic principle for describing the electron dynamics in curved nanotubes,” Uspekhi Fiz. Nauk 175(9), 1004–1010 (2005) [Soviet Phys. Uspekhi].
P. Duclos and P. Exner, “Curvature-induced bound states in quantum waveguides in two and three dimensions,” Rev. Math. Phys. 7(1), 73–102 (1995).
P. Exner and P. Šeba, “Bound states in curved quantum waveguides,” J. Math. Phys. 30(11), 2574–2580 (1989).
P. Exner, “Bound states in quantum waveguides of a slowly decaying curvature,” J. Math. Phys. 34(1), 23–28 (1993).
W. Bulla, F. Gesztesy, W. Renger, and B. Simon, “Weakly coupled bound states in quantum waveguides,” Proc. Amer. Math. Soc. 125(2), 1487–1495 (1997).
P. Exner and S. A. Vugalter, “Bounds states in a locally deformed waveguide: the critical value,” Lett. Math. Phys. 39(1), 59–68 (1997).
D. Borisov, P. Exner, R. Gadyl’shin, and D. Krejčiřík, “Bound states in a weakly deformed strips and layers,” Ann. Henri Poincaré 2(3), 553–572 (2001).
V. V. Grushin, “On the eigenvalues of finitely perturbed Laplace operators in infinite cylindrical domains,” Mat. Zametki 75(3), 360–371 (2004) [Math. Notes 75 (3–4), 331–340 (2004)].
V. V. Grushin, “Asymptotic behavior of the eigenvalues of the Schrödinger operator with transversal potential in a weakly curved infinite cylinder,” Mat. Zametki 77(5), 656–664 (2005) [Math. Notes 77 (5–6), 606–613 (2005)].
V. V. Grushin, “Asymptotic behavior of the eigenvalues of the Laplace operator in infinite cylinders perturbed by transverse extensions,” Mat. Zametki 81(3), 328–334 (2007) [Math. Notes 81 (3–4), 291–296 (2007)].
V. V. Grushin, “Asymptotic behavior of the eigenvalues of the Laplace operator in thin infinite tubes,” Mat. Zametki (in press).
R. R. Gadyl’shin, “On local perturbations of quantum waveguides,” Teoret. Mat. Fiz. 145(3), 358–371 (2005) [Theoret. and Math. Phys. 145 (3), 1678–1690 (2005)].
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1, in Travaux et recherches mathématiques, Vol. 18 (Dunod, Paris, 1968; Mir, Moscow, 1971).
L. Hörmander, Linear Partial Differential Operators, in Die Grundlehren der mathematischen Wissenschaften, Vol. 116 (Springer-Verlag, Berlin, 1963; Mir, Moscow, 1965).
L. I. Magarill and M. V. Entin, “Electrons in curvilinear quantum wire,” Zh. Éxper. Teoret. Fiz. 123(4), 867–876 (2003).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators (Academic Press, New York, 1978; Mir, Moscow, 1982).
M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic of general type,” Uspekhi Mat. Nauk 19(3), 53–161 (1964).
V. V. Grushin, “On a class of elliptic pseudodifferential operators degenerate on a submanifold,” Mat. Sb. 84(2), 163–195 (1971) [Math. USSR-Sb. 13, 155–185 (1971)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. V. Grushin, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 4, pp. 503–519.
Rights and permissions
About this article
Cite this article
Grushin, V.V. Asymptotic behavior of the eigenvalues of the Schrödinger operator in thin closed tubes. Math Notes 83, 463–477 (2008). https://doi.org/10.1134/S000143460803019X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143460803019X