May  2015, 14(3): 1239-1258. doi: 10.3934/cpaa.2015.14.1239

Spectral asymptotics of the Dirichlet Laplacian in a conical layer

1. 

IRMAR, Univ. Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France, France

2. 

BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo, 14 E48009 Bilbao, Basque Country, Spain

Published  March 2015

The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit.

On the one hand, we prove that, for any aperture, the eigenvalues accumulate below the threshold of the essential spectrum: For a small distance from the essential spectrum, the number of eigenvalues farther from the threshold than this distance behaves like the logarithm of the distance.

On the other hand, in the small aperture regime, we provide a two-term asymptotics of the first eigenvalues thanks to a priori localization estimates for the associated eigenfunctions. We prove that these eigenfunctions are localized in the conical cap at a scale of order the cubic root of the aperture angle anthat they get into the other part of the layer at a scale involving the logarithm of the aperture angle.
Citation: Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.

[2]

S. Agmon, Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of $N$-body Schrödinger Operators, vol. 29 of Mathematical Notes, Princeton University Press, Princeton, NJ, 1982.

[3]

S. Agmon, Bounds on exponential decay of eigenfunctions of Schrödinger operators, in Schrödinger Operators (Como, 1984), vol. 1159 of Lecture Notes in Math., Springer, Berlin, 1985, pp. 1-38. doi: 10.1007/BFb0080331.

[4]

J. Behrndt, P. Exner and V. Lotoreichik, Schrödinger operators with $\delta$-interactions supported on conical surfaces, J. Phys. A, (2014), submitted. doi: 10.1088/1751-8113/47/35/355202.

[5]

C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, vol. 3 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaïez.

[6]

M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Annalen der Physik, 389 (1927), 457-484.

[7]

G. Carron, P. Exner and D. Krejčiřík, Topologically nontrivial quantum layers, J. Math. Phys., 45 (2004), 774-784. doi: 10.1063/1.1635998.

[8]

J. Combes, P. Duclos and R. Seiler, The born-oppenheimer approximation, in Rigorous Atomic and Molecular Physics (G. Velo and A. Wightman eds.), vol. 74 of NATO Advanced Study Institutes Series, Springer US, 1981, 185-213.

[9]

H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, study ed., 1987.

[10]

M. Dauge, Y. Lafranche and N. Raymond, Quantum Waveguides with Corners, ESAIM: Proceedings, 35 (2012), 14-45. doi: 10.1051/proc/201235002.

[11]

M. Dauge and N. Raymond, Plane waveguides with corners in the small angle limit, J. Math. Phys., 53 (2012), 123529. doi: 10.1063/1.4769993.

[12]

M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, vol. 268 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511662195.

[13]

P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys., 7 (1995), 73-102. doi: 10.1142/S0129055X95000062.

[14]

P. Duclos, P. Exner, and D. Krejčiřík, Bound states in curved quantum layers, Comm. Math. Phys., 223 (2001), 13-28. doi: 10.1007/PL00005582.

[15]

P. Exner and P. Šeba, Bound states in curved quantum waveguides, J. Math. Phys., 30 (1989), 2574-2580. doi: 10.1063/1.528538.

[16]

P. Exner and M. Tater, Spectrum of Dirichlet Laplacian in a conical layer, J. Phys. A, 43 (2010), 474023. doi: 10.1088/1751-8113/43/47/474023.

[17]

P. Exner, P. Šeba and P. Št'oviček, On existence of a bound state in an L-shaped waveguide, Czechoslovak Journal of Physics, 39 (1989), 1181-1191.

[18]

J. Goldstone and R. L. Jaffe, Bound states in twisting tubes, Phys. Rev. B, 45 (1992), 14100-14107.

[19]

A. Hassell and S. Marshall, Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree -2, Trans. Am. Math. Soc., 360 (2008), 4145-4167. doi: 10.1090/S0002-9947-08-04479-6.

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, vol. 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988.

[21]

B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations, 9 (1984), 337-408. doi: 10.1080/03605308408820335.

[22]

B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Théor., 42 (1985), 127-212.

[23]

T. Jecko, On the mathematical treatment of the Born-Oppenheimer approximation, J. Math. Phys., 55, (2014) 053504. doi: 10.1063/1.4870855.

[24]

W. Kirsch and B. Simon, Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Physics, 183 (1988), 122-130. doi: 10.1016/0003-4916(88)90248-5.

[25]

M. Klein, A. Martinez, R. Seiler and X. P. Wang, On the Born-Oppenheimer expansion for polyatomic molecules, Comm. Math. Phys., 143 (1992), 607-639.

[26]

Y. Lafranche and D. Martin, Mélina++, bibliothèque de calculs éléments finis., http://anum-maths.univ-rennes1.fr/melina/, (2012).

[27]

A. Martinez, Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer, Ann. Inst. H. Poincaré Phys. Théor., 50 (1989), 239-257.

[28]

A. Martinez, A general effective Hamiltonian method, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 18 (2007), 269-277. doi: 10.4171/RLM/494.

[29]

A. Morame and F. Truc, Remarks on the spectrum of the Neumann problem with magnetic field in the half-space, J. Math. Phys., 46 (2005), 012105. doi: 10.1063/1.1827922.

[30]

S. Nazarov and A. Shanin, Trapped modes in angular joints of 2D waveguides, Appl. Anal., 93 (2014), 572-582. doi: 10.1080/00036811.2013.786046.

[31]

T. Ourmières-Bonafos, Dirichlet eigenvalues of cones in the small aperture limit, Journal of Spectral Theory, 4, Issue 3 (2014), 485?513. doi: 10.4171/JST/77.

[32]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.

[33]

B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.

[2]

S. Agmon, Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of $N$-body Schrödinger Operators, vol. 29 of Mathematical Notes, Princeton University Press, Princeton, NJ, 1982.

[3]

S. Agmon, Bounds on exponential decay of eigenfunctions of Schrödinger operators, in Schrödinger Operators (Como, 1984), vol. 1159 of Lecture Notes in Math., Springer, Berlin, 1985, pp. 1-38. doi: 10.1007/BFb0080331.

[4]

J. Behrndt, P. Exner and V. Lotoreichik, Schrödinger operators with $\delta$-interactions supported on conical surfaces, J. Phys. A, (2014), submitted. doi: 10.1088/1751-8113/47/35/355202.

[5]

C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, vol. 3 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaïez.

[6]

M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Annalen der Physik, 389 (1927), 457-484.

[7]

G. Carron, P. Exner and D. Krejčiřík, Topologically nontrivial quantum layers, J. Math. Phys., 45 (2004), 774-784. doi: 10.1063/1.1635998.

[8]

J. Combes, P. Duclos and R. Seiler, The born-oppenheimer approximation, in Rigorous Atomic and Molecular Physics (G. Velo and A. Wightman eds.), vol. 74 of NATO Advanced Study Institutes Series, Springer US, 1981, 185-213.

[9]

H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, study ed., 1987.

[10]

M. Dauge, Y. Lafranche and N. Raymond, Quantum Waveguides with Corners, ESAIM: Proceedings, 35 (2012), 14-45. doi: 10.1051/proc/201235002.

[11]

M. Dauge and N. Raymond, Plane waveguides with corners in the small angle limit, J. Math. Phys., 53 (2012), 123529. doi: 10.1063/1.4769993.

[12]

M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, vol. 268 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511662195.

[13]

P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys., 7 (1995), 73-102. doi: 10.1142/S0129055X95000062.

[14]

P. Duclos, P. Exner, and D. Krejčiřík, Bound states in curved quantum layers, Comm. Math. Phys., 223 (2001), 13-28. doi: 10.1007/PL00005582.

[15]

P. Exner and P. Šeba, Bound states in curved quantum waveguides, J. Math. Phys., 30 (1989), 2574-2580. doi: 10.1063/1.528538.

[16]

P. Exner and M. Tater, Spectrum of Dirichlet Laplacian in a conical layer, J. Phys. A, 43 (2010), 474023. doi: 10.1088/1751-8113/43/47/474023.

[17]

P. Exner, P. Šeba and P. Št'oviček, On existence of a bound state in an L-shaped waveguide, Czechoslovak Journal of Physics, 39 (1989), 1181-1191.

[18]

J. Goldstone and R. L. Jaffe, Bound states in twisting tubes, Phys. Rev. B, 45 (1992), 14100-14107.

[19]

A. Hassell and S. Marshall, Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree -2, Trans. Am. Math. Soc., 360 (2008), 4145-4167. doi: 10.1090/S0002-9947-08-04479-6.

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, vol. 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988.

[21]

B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations, 9 (1984), 337-408. doi: 10.1080/03605308408820335.

[22]

B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Théor., 42 (1985), 127-212.

[23]

T. Jecko, On the mathematical treatment of the Born-Oppenheimer approximation, J. Math. Phys., 55, (2014) 053504. doi: 10.1063/1.4870855.

[24]

W. Kirsch and B. Simon, Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Physics, 183 (1988), 122-130. doi: 10.1016/0003-4916(88)90248-5.

[25]

M. Klein, A. Martinez, R. Seiler and X. P. Wang, On the Born-Oppenheimer expansion for polyatomic molecules, Comm. Math. Phys., 143 (1992), 607-639.

[26]

Y. Lafranche and D. Martin, Mélina++, bibliothèque de calculs éléments finis., http://anum-maths.univ-rennes1.fr/melina/, (2012).

[27]

A. Martinez, Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer, Ann. Inst. H. Poincaré Phys. Théor., 50 (1989), 239-257.

[28]

A. Martinez, A general effective Hamiltonian method, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 18 (2007), 269-277. doi: 10.4171/RLM/494.

[29]

A. Morame and F. Truc, Remarks on the spectrum of the Neumann problem with magnetic field in the half-space, J. Math. Phys., 46 (2005), 012105. doi: 10.1063/1.1827922.

[30]

S. Nazarov and A. Shanin, Trapped modes in angular joints of 2D waveguides, Appl. Anal., 93 (2014), 572-582. doi: 10.1080/00036811.2013.786046.

[31]

T. Ourmières-Bonafos, Dirichlet eigenvalues of cones in the small aperture limit, Journal of Spectral Theory, 4, Issue 3 (2014), 485?513. doi: 10.4171/JST/77.

[32]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.

[33]

B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308.

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