Chenaud, B.; Duclos, P.; Freitas, P.; Krejčiřík, D. Geometrically induced discrete spectrum in curved tubes. (English) Zbl 1078.81022 Differ. Geom. Appl. 23, No. 2, 95-105 (2005). Summary: The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same cross-section and that the discrete spectrum is not empty. Cited in 42 Documents MSC: 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P05 General topics in linear spectral theory for PDEs 47F05 General theory of partial differential operators Keywords:quantum waveguides; bound states; Dirichlet Laplacian; Tang frame PDFBibTeX XMLCite \textit{B. Chenaud} et al., Differ. Geom. Appl. 23, No. 2, 95--105 (2005; Zbl 1078.81022) Full Text: DOI arXiv References: [1] Dermenjian, Y.; Durand, M.; Iftimie, V., Spectral analysis of an acoustic multistratified perturbed cylinder, Comm. Partial Differential Equations, 23, 1-2, 141-169 (1998) · Zbl 0907.47046 [2] Duclos, P.; Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys., 7, 73-102 (1995) · Zbl 0837.35037 [3] P. Exner, private communication, May 2003; P. Exner, private communication, May 2003 [4] Exner, P.; Freitas, P.; Krejčiřík, D., A lower bound to the spectral threshold in curved tubes, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 460, 2052, 3457-3467 (2004) · Zbl 1330.35360 [5] Exner, P.; Šeba, P., Bound states in curved quantum waveguides, J. Math. Phys., 30, 2574-2580 (1989) · Zbl 0693.46066 [6] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0691.35001 [7] Goldstone, J.; Jaffe, R. L., Bound states in twisting tubes, Phys. Rev. B, 45, 14100-14107 (1992) [8] Klingenberg, W., A Course in Differential Geometry (1978), Springer: Springer New York [9] Krejčiřík, D.; Tiedra de Aldecoa, R., The nature of the essential spectrum in curved quantum waveguides, J. Phys. A, 37, 20, 5449-5466 (2004) · Zbl 1062.81046 [10] Krejčiřík, D.; Kříž, J., On the spectrum of curved quantum waveguides, Publ. RIMS Kyoto Univ., 41, 3 (2005) · Zbl 1113.35143 [11] Kurzweil, J., Ordinary Differential Equations (1986), Elsevier: Elsevier Amsterdam · Zbl 0619.26006 [12] Londergan, J. T.; Carini, J. P.; Murdock, D. P., Binding and Scattering in Two-dimensional Systems, LNP, vol. 60 (1999), Springer: Springer Berlin · Zbl 0997.81511 [13] Renger, W.; Bulla, W., Existence of bound states in quantum waveguides under weak conditions, Lett. Math. Phys., 35, 1-12 (1995) · Zbl 0838.35087 [14] Spivak, M., A Comprehensive Introduction to Differential Geometry, vol. II (1979), Publish or Perish: Publish or Perish Houston, TX · Zbl 0439.53002 [15] Tsao, C. Y.H.; Gambling, W. A., Curvilinear optical fibre waveguide: characterization of bound modes and radiative field, Proc. R. Soc. Lond. A, 425, 1-16 (1989) [16] Weidmann, J., Linear Operators in Hilbert Spaces (1980), Springer: Springer New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.