Resonances near thresholds in slightly twisted waveguides

Bruneau, Vincent; Miranda, Pablo; Popoff, Nicolas

doi:10.1090/proc/14141

Resonances near thresholds in slightly twisted waveguides

Authors: Vincent Bruneau, Pablo Miranda and Nicolas Popoff
Journal: Proc. Amer. Math. Soc. 146 (2018), 4801-4812
MSC (2010): Primary 35J10, 81Q10, 35P20
DOI: https://doi.org/10.1090/proc/14141
Published electronically: July 23, 2018
MathSciNet review: 3856147
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator and show the existence of exactly one pole near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior.

[1] D. Bollé, F. Gesztesy, and S. F. J. Wilk, A complete treatment of low-energy scattering in one dimension, J. Operator Theory 13 (1985), no. 1, 3–31. MR 768299
[2] Jean-François Bony, Vincent Bruneau, and Georgi Raikov, Resonances and spectral shift function near the Landau levels, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 629–671 (English, with English and French summaries). MR 2310953
[3] Jean-Marc Bouclet and Vincent Bruneau, Semiclassical resonances of Schrödinger operators as zeroes of regularized determinants, Int. Math. Res. Not. IMRN 7 (2008), Art. ID rnn002, 55. MR 2428303, https://doi.org/10.1093/imrn/rnn002
[4] Philippe Briet, Hynek Kovařík, Georgi Raikov, and Eric Soccorsi, Eigenvalue asymptotics in a twisted waveguide, Comm. Partial Differential Equations 34 (2009), no. 7-9, 818–836. MR 2560302, https://doi.org/10.1080/03605300902892337
[5] W. Bulla, F. Gesztesy, W. Renger, and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1487–1495. MR 1371117, https://doi.org/10.1090/S0002-9939-97-03726-X
[6] P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), no. 1, 73–102. MR 1310767, https://doi.org/10.1142/S0129055X95000062
[7] P. Duclos, P. Exner, and B. Meller, Exponential bounds on curvature-induced resonances in a two-dimensional Dirichlet tube, Helv. Phys. Acta 71 (1998), no. 2, 133–162. MR 1609179
[8] T. Ekholm, H. Kovařík, and D. Krejčiřík, A Hardy inequality in twisted waveguides, Arch. Ration. Mech. Anal. 188 (2008), no. 2, 245–264. MR 2385742, https://doi.org/10.1007/s00205-007-0106-0
[9] V. V. Grushin, On the eigenvalues of a finitely perturbed Laplace operator in infinite cylindrical domains, Mat. Zametki 75 (2004), no. 3, 360–371 (Russian, with Russian summary); English transl., Math. Notes 75 (2004), no. 3-4, 331–340. MR 2068799, https://doi.org/10.1023/B:MATN.0000023312.41107.72
[10] Evans M. Harrell II, Perturbation theory and atomic resonances since Schrödinger’s time, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 227–248. MR 2310205, https://doi.org/10.1090/pspum/076.1/2310205
[11] Hynek Kovařík and Andrea Sacchetti, Resonances in twisted quantum waveguides, J. Phys. A 40 (2007), no. 29, 8371–8384. MR 2371239, https://doi.org/10.1088/1751-8113/40/29/012
[12] David Krejčiřík, Twisting versus bending in quantum waveguides, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 617–637. MR 2459893, https://doi.org/10.1090/pspum/077/2459893
[13] L. Nedelec, Sur les résonances de l’opérateur de Dirichlet dans un tube, Comm. Partial Differential Equations 22 (1997), no. 1-2, 143–163 (French, with English summary). MR 1434141, https://doi.org/10.1080/03605309708821258
[14] Barry Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288. MR 404846, https://doi.org/10.1016/0003-4916(76)90038-5
[15] Barry Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), no. 2, 396–420. MR 1802901, https://doi.org/10.1006/jfan.2000.3669
[16] Johannes Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, Mém. Soc. Math. Fr. (N.S.) 136 (2014), vi+144 (English, with English and French summaries). MR 3288114, https://doi.org/10.24033/msmf.446