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Microlocal Methods in Mathematical Physics and Global Analysis pp 29–32Cite as

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The Effective Hamiltonian in Curved Quantum Waveguides and When It Does Not Work

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Abstract

We are concerned with the singular operator limit for the Dirichlet Laplacian in a three-dimensional curved tube (cf. Fig. 1) when its cross-section shrinks to zero.

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References

  1. R. L. Bishop, There is more than one way to frame a curve, The American Mathematical Monthly 82 (1975), 246–251.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Bouchitté, M. L. Mascarenhas, and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides, ESAIM: Control, Optimisation and Calculus of Variations 13 (2007), 793–808.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. R. de Oliveira, Quantum singular operator limits of thin Dirichlet tubes via \(\Gamma \) -convergence, Rep. Math. Phys. 66 (2010) 375–406.

    Google Scholar 

  5. D. Krejčiřík, Twisting versus bending in quantum waveguides, Analysis on Graphs and its Applications, Cambridge, 2007 (P. Exner et al., ed.), Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 617–636; arXiv:0712.3371 [math-ph].

    Google Scholar 

  6. J. Lampart, S. Teufel, and J. Wachsmuth, Effective Hamiltonians for thin Dirichlet tubes with varying cross-section, Mathematical Results in Quantum Physics, September, 2010, xi + 274 p.; World Scientific, Singapore, 2011, pp. 183–189; arXiv:1011.3645 [math-ph].

    Google Scholar 

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Author information

Authors and Affiliations

  1. Nuclear Physics Institute ASCR, 25068, Řež, Czech Republic

    David Krejčiřík

  2. Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Břehová 7, 11519, Praha 1, Czech Republic

    Helena Šediváková

Authors
  1. David Krejčiřík
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  2. Helena Šediváková
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Corresponding author

Correspondence to David Krejčiřík .

Editor information

Editors and Affiliations

  1. Ammerländer Heerstr. 114-118, Oldenburg, 26111, Germany

    Daniel Grieser

  2. Inst. Mathematik, Universität Tübingen, Auf der Morgenstelle 10, Tübingen, 72076, Germany

    Stefan Teufel

  3. , Department of Mathematics, Bldg. 380, Stanford University, 450, Serra Mall, Stanford, 94305-2125, California, USA

    Andras Vasy

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Krejčiřík, D., Šediváková, H. (2013). The Effective Hamiltonian in Curved Quantum Waveguides and When It Does Not Work. In: Grieser, D., Teufel, S., Vasy, A. (eds) Microlocal Methods in Mathematical Physics and Global Analysis. Trends in Mathematics(). Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0466-0_7

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