×

Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary. (English) Zbl 1380.65380

Summary: A numerical scheme to compute the spectrum of a large class of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary in any dimension is presented. The algorithm is based on the characterisation of a large class of self-adjoint extensions of Laplace-Beltrami operators in terms of their associated quadratic forms. The convergence of the scheme is proved. A two-dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R01 PDEs on manifolds

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, R. A.; Fournier, J. J.F., Sobolev Spaces, Pure Appl. Math. (2003), Academic Press: Academic Press Oxford · Zbl 1098.46001
[2] Asorey, M.; Alvarez, G. G.; Muñoz-Castañeda, J. M., Casimir effect and global theory of boundary conditions, J. Phys. A, Math. Gen., 39, 21, 6127 (2006) · Zbl 1094.81035
[3] Asorey, M.; Balachandran, A. P.; Pérez-Pardo, J. M., Edge states: topological insulators, superconductors and QCD chiral bags, J. High Energy Phys., 2013, 12, Article 073 pp. (2013)
[4] Asorey, M.; Balachandran, A. P.; Pérez-Pardo, J. M., Edge states at phase boundaries and their stability, Rev. Math. Phys., 28, 9, Article 1650020 pp. (2016) · Zbl 1351.81079
[5] Asorey, M.; Esteve, J. G.; Pacheco, A. F., Planar rotor: the \(θ\)-vacuum structure, and some approximate methods in quantum mechanics, Phys. Rev. D, 27, 1852-1868 (1982)
[6] Asorey, M.; Ibort, A.; Marmo, G., The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators, Int. J. Geom. Methods Mod. Phys., 12, 6, Article 1561007 pp. (2015) · Zbl 1331.58021
[7] Babuška, I.; Nobile, F.; Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45, 3, 1005-1034 (2007) · Zbl 1151.65008
[8] Babuška, I.; Osborn, J., Eigenvalue Problems Handbook of Numerical Analysis, vol. II (1991), Elsevier: Elsevier North-Holland
[9] Babuška, I.; Suri, M., On locking and robustness in the finite element method, SIAM J. Numer. Anal., 29, 1261-1293 (1992) · Zbl 0763.65085
[10] Babuška, I.; Tempone, R.; Zouraris, G. E., Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42, 800-825 (2004) · Zbl 1080.65003
[11] Babuška, I.; Tempone, R.; Zouraris, G. E., Solving elliptic boundary value problems with uncertain coefficients by the finite element method - the stochastic formulation, Comput. Methods Appl. Mech. Eng., 194, 1251-1294 (2005) · Zbl 1087.65004
[12] Bernevig, B. A.; Zhang, S.-C., Quantum spin hall effect, Phys. Rev. Lett., 96, 10, Article 106802 pp. (2006)
[13] Boffi, D., Finite element approximation of eigenvalue problems, Acta Numer., 19, 1-120 (2010) · Zbl 1242.65110
[14] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (2008), Springer · Zbl 1135.65042
[15] Carlone, R.; Posilicano, A., A quantum hybrid with a thin antenna at the vertex of a wedge, Phys. Lett. A, 381, 12, 1076-1080 (2017) · Zbl 1372.81068
[16] Clarke, J.; Wilhelm, F. K., Superconducting quantum bits, Nature, 453, 1031-1042 (2008)
[17] Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E., Electronic transport in two dimensional graphene, Rev. Mod. Phys., 83, 407-470 (2011)
[18] Davies, E. B., Spectral Theory and Differential Operators (1995), Cambridge University Press · Zbl 0893.47004
[19] Dell’Antonio, G. F.; Figari, R.; Teta, A., A Brief Review on Point Interactions in Inverse Problems and Imaging, Lect. Notes Math. (2008), Springer
[20] Demmel, J. W., Applied Numerical Linear Algebra (1997), SIAM · Zbl 0879.65017
[21] Exner, P.; Šeba, P., Resonance statistics in a microwave cavity with a thin antenna, Phys. Lett. A, 228, 3, 146-150 (1997) · Zbl 0962.78501
[22] Exner, P.; Šeba, P., A “hybrid plane” with spin-orbit interaction, Russ. J. Math. Phys., 14, 4, 430-434 (2007) · Zbl 1175.81092
[23] Grubb, G., A characterization of the nonlocal boundary value problems associated with an elliptic operator, Ann. Sc. Norm. Super., 22, 3, 425-513 (1968) · Zbl 0182.14501
[24] Grubb, G., Krein-like extensions and the lower boundedness problem for elliptic operators, J. Differ. Equ., 252, 852-885 (2012) · Zbl 1242.47041
[25] Hahn, D. W.; Necati Özisik, M., Heat Conduction (2012), John Wiley & Sons
[26] Ibort, A.; Lledó, F.; Pérez-Pardo, J. M., On self-adjoint extensions and symmetries in quantum mechanics, Ann. Inst. Henri Poincaré, 16, 2367-2397 (2015) · Zbl 1456.81179
[27] Ibort, A.; Lledó, F.; Pérez-Pardo, J. M., Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary, J. Funct. Anal., 268, 3, 634-670 (2015) · Zbl 1310.47032
[28] Ibort, A.; Marmo, G.; Pérez-Pardo, J. M., Boundary dynamics driven entanglement, J. Phys. A, Math. Theor., 47, 38, Article 385301 pp. (2014) · Zbl 1298.81016
[29] Ibort, A.; Pérez-Pardo, J. M., Numerical solutions of the spectral problem for arbitrary self-adjoint extensions of 1D Schrödinger equation, SIAM J. Numer. Anal., 51, 2, 1254-1279 (2013) · Zbl 1279.65093
[30] Ibort, A.; Pérez-Pardo, J. M., On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics, Int. J. Geom. Methods Mod. Phys., 12, 6, Article 1560005 pp. (2015) · Zbl 1327.81199
[31] Kato, T., Perturbation Theory for Linear Operators, Class. Math. (1995), Springer · Zbl 0836.47009
[32] Kittel, C., Introduction to Solid State Physics (1996), John Wiley & Sons: John Wiley & Sons New York
[33] Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Grundlehren Math. Wiss., vol. I (1972), Springer · Zbl 0227.35001
[34] Marolf, D., Interpolating between topologies: Casimir energies, Phys. Lett. B, 392, 287 (1997)
[35] Martinis, J. M., Decoherence in Josephson qubits from dielectric loss, Phys. Rev. Lett., 95, Article 210503 pp. (2005)
[36] Morandi, G., Quantum Hall Effect: Topological Problems in Condensed-Matter Physics, Monogr. Textb. Phys. Sci., Lect. Notes, vol. 10 (1988), Bibliopolis: Bibliopolis Naples · Zbl 0708.53058
[37] Mohsen, A., On the impedance boundary conditions, Appl. Math. Model., 6, 5, 405-407 (1982)
[38] Norton, R., Numerical Computation of Band Gaps in Photonic Crystal Fibres (2008), University of Bath, PhD Thesis
[39] Pérez-Pardo, J. M., Dirac-like operators on the Hilbert space of differential forms on manifolds with boundaries, Int. J. Geom. Methods Mod. Phys., 14 (2017) · Zbl 1375.35434
[40] Pérez-Pardo, J. M.; Barbero-Liñán, M.; Ibort, A., Boundary dynamics and topology change in quantum mechanics, Int. J. Geom. Methods Mod. Phys., 12, Article 1560011 pp. (2015) · Zbl 1332.35315
[41] Plunien, G.; Müller, B.; Greiner, W., The Casimir effect, Phys. Rep., 134, 87-193 (1986)
[42] Ram-Mohan, R., Finite Element and Boundary Element Applications in Quantum Mechanics (2002), Oxford University Press
[43] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, volume I (1972), Academic Press: Academic Press New York
[44] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, volume II (1975), Academic Press: Academic Press New York
[45] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, volume IV (1978), Academic Press: Academic Press New York
[46] Strouboulis, T.; Babuska, I.; Copps, K., The design and analysis of the generalised finite element method, Comput. Methods Appl. Mech. Eng., 181, 43-69 (2000) · Zbl 0983.65127
[47] Sukumar, N.; Pask, J. E., Classical and enriched finite element formulations for Bloch-periodic boundary conditions, Int. J. Numer. Methods Eng., 77, 1121-1138 (2009) · Zbl 1156.81313
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.