S. Agmon, Lectures on Exponential Decay of Solution of Second-Order Elliptic Equation, 1982.
DOI : 10.1515/9781400853076

. Forchel, Whispering gallery mode lasing in electrically driven quantum dot micropillars

G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Commentarii Mathematici Helvetici, vol.69, issue.1, pp.142-154, 1994.
DOI : 10.1007/BF02564478

F. A. Alhargan, A Complete Method for the Computations of Mathieu characteristic numbers of integer orders, SIAM Rev, pp.239-255, 1996.

P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Physical Review, vol.109, issue.5, p.1492, 1958.
DOI : 10.1103/PhysRev.109.1492

J. M. Arrieta and D. Krejcirik, Geometric versus spectral convergence for the Neumann Laplacian under exterior perturbations of the domain, Integral methods in science and engineering, pp.9-19, 2010.

J. M. Arrieta, Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case, Transactions of the American Mathematical Society, vol.347, issue.9, pp.3503-3531, 1995.
DOI : 10.1090/S0002-9947-1995-1297521-1

J. M. Arrieta, Neumann Eigenvalue Problems on Exterior Perturbations of the Domain, Journal of Differential Equations, vol.118, issue.1, pp.54-103, 1995.
DOI : 10.1006/jdeq.1995.1067

M. S. Ashbaugh and P. Exner, Lower bounds to bound state energies in bent tubes, Physics Letters A, vol.150, issue.3-4, pp.183-186, 1990.
DOI : 10.1016/0375-9601(90)90118-8

R. Aurich and F. Steiner, Exact theory for the quantum eigenstates of a strongly chaotic system, Physica D: Nonlinear Phenomena, vol.48, issue.2-3, pp.445-470, 1991.
DOI : 10.1016/0167-2789(91)90097-S

Y. Avishai, D. Bessis, B. G. Giraud, and G. Mantica, Quantum bound states in open geometries, Physical Review B, vol.44, issue.15, pp.8028-8034, 1991.
DOI : 10.1103/PhysRevB.44.8028

V. M. Babich and V. F. Lazutkin, Eigenfunctions Concentrated Near a Closed Geodesic, Consultant's Bureau, pp.9-18, 1968.
DOI : 10.1007/978-1-4684-7592-0_2

A. Bäcker, R. Schubert, and P. Stifter, On the number of bouncing ball modes in billiards, Journal of Physics A: Mathematical and General, vol.30, issue.19, pp.6783-6795, 1997.
DOI : 10.1088/0305-4470/30/19/017

A. Bäcker and R. Schubert, Rate of quantum ergodicity in Euclidean billiards, Physical Review E, vol.57, issue.5, 1998.
DOI : 10.1103/PhysRevE.57.5425

B. Balagurov, . Ya, and V. G. Vaks, Random walks of a particle on lattices with traps, J. Exper. Theor. Phys, vol.38, p.968, 1974.

A. H. Barnett, Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards, Communications on Pure and Applied Mathematics, vol.175, issue.10, pp.1457-1488, 2006.
DOI : 10.1002/cpa.20150

J. T. Beale, Scattering frequencies of resonators, Communications on Pure and Applied Mathematics, vol.37, issue.4, pp.549-564, 1973.
DOI : 10.1002/cpa.3160260408

D. Belitz and T. R. Kirkpatrick, The Anderson-Mott transition, Reviews of Modern Physics, vol.66, issue.2, p.261, 1994.
DOI : 10.1103/RevModPhys.66.261

O. Bénichou, D. S. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Optimal Reaction Time for Surface-Mediated Diffusion, Physical Review Letters, vol.105, issue.15, p.150606, 2010.
DOI : 10.1103/PhysRevLett.105.150606

O. Bénichou, D. S. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Mean First-Passage Time of Surface-Mediated Diffusion in Spherical Domains, Journal of Statistical Physics, vol.122, issue.4, pp.657-685, 2011.
DOI : 10.1007/s10955-011-0138-6

O. Bénichou and R. Voituriez, Narrow-Escape Time Problem: Time Needed for a Particle to Exit a Confining Domain through a Small Window, Physical Review Letters, vol.100, issue.16, p.168105, 2008.
DOI : 10.1103/PhysRevLett.100.168105

T. Berry, S. M. Heilman, and R. S. Strichartz, Outer Approximation of the Spectrum of a Fractal Laplacian, Experimental Mathematics, vol.18, issue.4, pp.449-480, 2009.
DOI : 10.1080/10586458.2009.10129061

W. Bies, L. Kaplan, M. Haggerty, and E. Heller, Localization of eigenfunctions in the stadium billiard, Physical Review E, vol.63, issue.6, 2001.
DOI : 10.1103/PhysRevE.63.066214

D. A. Bini, L. Gemignani, and F. Tisseur, The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem, SIAM Journal on Matrix Analysis and Applications, vol.27, issue.1, pp.153-175, 2005.
DOI : 10.1137/S0895479803429788

M. S. Birman, O spektre singulyarnix granichnix zadach, Math. Sb, vol.55, pp.125-174, 1961.

A. S. Bonnet-ben, P. Dhia, and . Joly, Mathematical analysis of guided water waves, SIAM J. Appl. Math, vol.53, pp.1507-1550, 1993.
URL : https://hal.archives-ouvertes.fr/inria-00074932

F. Bowman, Introduction to Bessel functions, 1958.

R. Brown, P. D. Hislop, and A. Martinez, Eigenvalues and Resonances for Domains with Tubes: Neumann Boundary Conditions, Journal of Differential Equations, vol.115, issue.2, pp.458-476, 1995.
DOI : 10.1006/jdeq.1995.1023

K. R. Brownstein and C. E. Tarr, Importance of classical diffusion in NMR studies of water in biological cells, Physical Review A, vol.19, issue.6, pp.2446-2453, 1979.
DOI : 10.1103/PhysRevA.19.2446

E. N. Bulgakov, P. Exner, K. N. Pichugin, and A. F. Sadreev, Multiple bound states in scissor-shaped waveguides, Physical Review B, vol.66, issue.15, p.155109, 2002.
DOI : 10.1103/PhysRevB.66.155109

W. Bulla, F. Gesztesy, W. Renger, and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer, pp.1487-1495, 1997.

J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numerische Mathematik, vol.13, issue.1, pp.31-48, 1978.
DOI : 10.1007/BF01396012

N. Burq and M. Zworski, Bouncing Ball Modes and Quantum Chaos, SIAM Review, vol.47, issue.1, pp.43-49, 2005.
DOI : 10.1137/S0036144503429248

URL : http://arxiv.org/abs/math/0306278

J. P. Carini, J. T. Londergan, K. Mullen, and D. P. Murdock, Bound states and resonances in waveguides and quantum wires, Physical Review B, vol.46, issue.23, pp.15538-15541, 1992.
DOI : 10.1103/PhysRevB.46.15538

J. P. Carini, J. T. Londergan, K. Mullen, and D. P. Murdock, Multiple bound states in sharply bent waveguides, Multiple bound states in sharply bent waveguides, pp.4503-4515, 1993.
DOI : 10.1103/PhysRevB.48.4503

J. P. Carini, J. T. Londergan, D. P. Murdock, D. Trinkle, and C. S. Yung, Bound states in waveguides and bent quantum wires. I. Applications to waveguide systems, Physical Review B, vol.55, issue.15, p.9842, 1997.
DOI : 10.1103/PhysRevB.55.9842

J. Carrier, L. Greengard, and V. Rokhlin, A Fast Adaptive Multipole Algorithm for Particle Simulations, SIAM Journal on Scientific and Statistical Computing, vol.9, issue.4, pp.669-686, 1988.
DOI : 10.1137/0909044

L. G. Chambers, An upper bound for the first zero of Bessel functions, Mathematics of Computation, vol.38, issue.158, pp.589-591, 1982.
DOI : 10.1090/S0025-5718-1982-0645673-0

G. Chen, P. J. Morris, and J. Zhou, Visualization of special eigenmodes shapes of a vibrating elliptical membrane, SIAM Rev, pp.453-469, 1994.

B. Chenaud, P. Duclos, P. Freitas, and D. Krejcirík, Geometrically induced discrete spectrum in curved tubes, Differential Geometry and its Applications, vol.23, issue.2, pp.95-105, 2005.
DOI : 10.1016/j.difgeo.2005.05.001

A. F. Cheviakov, M. J. Ward, and . Straube, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere, Multiscale Modeling & Simulation, vol.8, issue.3, pp.836-870, 2010.
DOI : 10.1137/100782620

A. Cheviakov and M. J. Ward, Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps, Mathematical and Computer Modelling, vol.53, issue.7-8, pp.7-8, 2011.
DOI : 10.1016/j.mcm.2010.02.025

Y. Colin-de-verdière, Ergodicit??? et fonctions propres du laplacien, Communications in Mathematical Physics, vol.29, issue.3, pp.497-502, 1985.
DOI : 10.1007/BF01209296

S. Condamin, O. Bénichou, and M. Moreau, First-Passage Times for Random Walks in Bounded Domains, Physical Review Letters, vol.95, issue.26, p.260601, 2005.
DOI : 10.1103/PhysRevLett.95.260601

S. Condamin, O. Bénichou, and M. Moreau, First-exit times and residence times for discrete random walks on finite lattices, Physical Review E, vol.72, issue.1, p.16127, 2005.
DOI : 10.1103/PhysRevE.72.016127

S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature, vol.2, issue.7166, p.77, 2007.
DOI : 10.1038/nature06201

S. Condamin, O. Bénichou, and M. Moreau, Random walks and Brownian motion: A method of computation for first-passage times and related quantities in confined geometries, Physical Review E, vol.75, issue.2, p.21111, 2007.
DOI : 10.1103/PhysRevE.75.021111

S. Condamin, V. Tejedor, and O. Bénichou, Occupation times of random walks in confined geometries: From random trap model to diffusion-limited reactions, Physical Review E, vol.76, issue.5, p.50102, 2007.
DOI : 10.1103/PhysRevE.76.050102

M. Conti, S. Terracini, and G. Veizini, An optimal partition problem related to nonlinear eigenvalues, Journal of Functional Analysis, vol.198, issue.1, pp.160-196, 2003.
DOI : 10.1016/S0022-1236(02)00105-2

M. Conti, S. Terracini, and G. Veizini, On a class of optimal partition problems related to the Fu????k spectrum and to the monotonicity formulae, Calculus of Variations, vol.134, issue.1, pp.45-72, 2005.
DOI : 10.1007/s00526-004-0266-9

M. Coppens, The effect of fractal surface roughness on diffusion and reaction in porous catalysts ??? from fundamentals to practical applications, Catalysis Today, vol.53, issue.2, pp.225-243, 1999.
DOI : 10.1016/S0920-5861(99)00118-2

J. Crank, The Mathematics of Diffusion, 1975.

D. Daners, Dirichlet problems on varying domains, Journal of Differential Equations, vol.188, issue.2, pp.591-624, 2003.
DOI : 10.1016/S0022-0396(02)00105-5

URL : http://doi.org/10.1016/s0022-0396(02)00105-5

B. Daudert and M. Lapidus, LOCALIZATION ON SNOWFLAKE DOMAINS, Fractals, vol.15, issue.03, p.255, 2007.
DOI : 10.1142/S0218348X0700354X

E. B. Davies and L. Parnovski, Trapped modes in acoustic waveguides, The Quarterly Journal of Mechanics and Applied Mathematics, vol.51, issue.3, pp.477-492, 1998.
DOI : 10.1093/qjmam/51.3.477

A. Delitsyn, B. T. Nguyen, and D. S. Grebenkov, Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles, The European Physical Journal B, vol.69, issue.11, p.371, 2012.
DOI : 10.1140/epjb/e2012-30286-8

A. Delitsyn, B. T. Nguyen, and D. S. Grebenkov, Trapped modes in finite quatum waveguides, European Physical Journal B, vol.85, issue.6, 2012.

A. L. Delitsyn, The Discrete Spectrum of the Laplace Operator in a Cylinder with Locally Perturbed Boundary, Differential Equations, vol.40, issue.2, pp.207-217, 2004.
DOI : 10.1023/B:DIEQ.0000033710.40234.3c

J. Dittrich and J. Kríz, Curved planar quantum wires with Dirichlet and Neumann boundary conditions, Journal of Physics A: Mathematical and General, vol.35, issue.20, pp.269-275, 2002.
DOI : 10.1088/0305-4470/35/20/101

H. Donnelly, Quantum unique ergodicity, Proc. Amer, pp.2945-2951, 2003.

H. Donnelly, Spectral gap for convex planar domains, Mathematische Zeitschrift, vol.12, issue.1-2, pp.1-3, 2011.
DOI : 10.1007/s00209-009-0629-1

P. Duclos and P. Exner, CURVATURE-INDUCED BOUND STATES IN QUANTUM WAVEGUIDES IN TWO AND THREE DIMENSIONS, Reviews in Mathematical Physics, vol.07, issue.01, pp.73-102, 1995.
DOI : 10.1142/S0129055X95000062

A. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, Journal of Computational and Applied Mathematics, vol.133, issue.1-2, pp.65-83, 2001.
DOI : 10.1016/S0377-0427(00)00635-X

D. V. Evans, Trapped acoustic modes, IMA Journal of Applied Mathematics, vol.49, issue.1, pp.45-60, 1992.
DOI : 10.1093/imamat/49.1.45

D. V. Evans, M. Levitin, and D. Vassiliev, Existence theorems for trapped modes, Journal of Fluid Mechanics, vol.100, issue.-1, pp.21-31, 1994.
DOI : 10.1016/0022-0248(90)90248-J

URL : http://discovery.ucl.ac.uk/170542/1/download15.pdf

C. Even, S. Russ, V. Repain, P. Pieranski, and B. Sapoval, Localizations in Fractal Drums: An Experimental Study, Physical Review Letters, vol.83, issue.4, pp.83-726, 1999.
DOI : 10.1103/PhysRevLett.83.726

F. Evers, A. D. Mirlin, and A. Transitions, Anderson transitions, Reviews of Modern Physics, vol.80, issue.4, p.1355, 2008.
DOI : 10.1103/RevModPhys.80.1355

P. Exner, P. Freitas, and D. Krej?i?ík, A lower bound to the spectral threshold in curved tubes, Proc. R. Soc. Lond. A, pp.3457-3467, 2004.
DOI : 10.1098/rspa.2004.1356

P. Exner and P. Seba, Bound states in curved quantum waveguides, Journal of Mathematical Physics, vol.30, issue.11, pp.2574-2580, 1989.
DOI : 10.1063/1.528538

P. Exner, P. Seba, M. Tater, and D. Vanek, Bound states and scattering in quantum waveguides coupled laterally through a boundary window, Journal of Mathematical Physics, vol.37, issue.10, pp.37-4867, 1996.
DOI : 10.1063/1.531673

S. Felix, M. Asch, M. Filoche, and B. Sapoval, Localization and increased damping in irregular acoustic cavities, Journal of Sound and Vibration, vol.299, issue.4-5, pp.299-965, 2007.
DOI : 10.1016/j.jsv.2006.07.036

M. Filoche and S. Mayboroda, Strong Localization Induced by One Clamped Point in Thin Plate Vibrations, Physical Review Letters, vol.103, issue.25, p.254301, 2009.
DOI : 10.1103/PhysRevLett.103.254301

M. Filoche and S. Mayboroda, The Hidden Landscape of Localization, pp.1107-0397, 2011.

P. Freitas and D. Krejcirik, Unbounded planar domains whose second nodal line does not touch the boundary, Mathematical Research Letters, vol.14, issue.1, pp.107-111, 2007.
DOI : 10.4310/MRL.2007.v14.n1.a9

P. Freitas and D. Krej?i?ík, Waveguides with Combined Dirichlet and Robin Boundary Conditions, Mathematical Physics, Analysis and Geometry, vol.219, issue.3, pp.335-352, 2006.
DOI : 10.1007/s11040-007-9015-6

P. Freitas and D. Krej?i?ík, A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains, Proc. Amer, pp.2997-3006, 2008.
DOI : 10.1090/S0002-9939-08-09399-4

D. Frenkel and R. Portugal, Algebraic methods to compute Mathieu functions, Journal of Physics A: Mathematical and General, vol.34, issue.17
DOI : 10.1088/0305-4470/34/17/302

M. Gu and S. C. , A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem, SIAM Journal on Matrix Analysis and Applications, vol.15, issue.4, pp.1266-1276, 1992.
DOI : 10.1137/S089547989223924X

M. Gu and S. C. , A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem, SIAM Journal on Matrix Analysis and Applications, vol.16, issue.1, pp.172-191, 1995.
DOI : 10.1137/S0895479892241287

P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math, J, vol.71, pp.559-607, 1993.

I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, 1963.

J. Goldstone and R. L. Jaffe, Bound states in twisting tubes, Physical Review B, vol.45, issue.24, pp.14100-14107, 1992.
DOI : 10.1103/PhysRevB.45.14100

L. Greengard and V. , A fast algorithm for particle simulations, Journal of Computational Physics, vol.73, issue.2, pp.325-348, 1987.
DOI : 10.1016/0021-9991(87)90140-9

P. Grassberger and I. Procaccia, The long time properties of diffusion in a medium with static traps, The Journal of Chemical Physics, vol.77, issue.12, pp.6281-6284, 1982.
DOI : 10.1063/1.443832

D. S. Grebenkov and B. T. Nguyen, Geometrical Structure of Laplacian Eigenfunctions, SIAM Review, vol.55, issue.4
DOI : 10.1137/120880173

D. S. Grebenkov, Multiple correlation function approach: rigorous results for simples geometries, Diff. Fundam, vol.5, issue.1, 2007.

D. S. Grebenkov, Residence times and other functionals of reflected Brownian motion, Physical Review E, vol.76, issue.4, p.41139, 2007.
DOI : 10.1103/PhysRevE.76.041139

D. S. Grebenkov, NMR survey of reflected Brownian motion, Reviews of Modern Physics, vol.79, issue.3, pp.1077-1137, 2007.
DOI : 10.1103/RevModPhys.79.1077

D. S. Grebenkov, Laplacian eigenfunctions in NMR I. A numerical tool, Concepts Magn, Reson. A, vol.32, pp.277-301, 2008.

D. S. Grebenkov, Analytical solution for restricted diffusion in circular and spherical layers under inhomogeneous magnetic fields, The Journal of Chemical Physics, vol.128, issue.13, p.134702, 2008.
DOI : 10.1063/1.2841367

D. S. Grebenkov, Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary, Physical Review E, vol.81, issue.2, p.21128, 2010.
DOI : 10.1103/PhysRevE.81.021128

P. Grisvard, Elliptic Problem for nonsmooth domain, 1985.
DOI : 10.1137/1.9781611972030

O. Haeberle, B. Sapoval, K. Menou, and H. Vach, Observation of vibrational modes of irregular drums, Applied Physics Letters, vol.73, issue.23, pp.73-3357, 1998.
DOI : 10.1063/1.122768

URL : https://hal.archives-ouvertes.fr/hal-00958099

A. Hassell, Ergodic billiards that are not quantum unique ergodic, Annals of Mathematics, vol.171, issue.1, pp.605-618, 2010.
DOI : 10.4007/annals.2010.171.605

URL : https://hal.archives-ouvertes.fr/hal-00345657

B. Hébert, B. Sapoval, and S. Russ, Experimental study of a fractal acoustical cavity, The Journal of the Acoustical Society of America, vol.105, issue.3, p.1567, 1999.
DOI : 10.1121/1.426696

S. M. Heilman and R. S. Strichartz, Localized Eigenfunctions: Here You See Them, There You Don't, Notices Amer, Math. Soc, vol.57, pp.624-629, 2010.

B. Helffer and T. Hoffmann-ostenhof, Converse spectral problems for nodal domains, Mos. Math. Journal, vol.7, issue.1, pp.67-84, 2007.

B. Helffer, T. Hoffmann-ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, issue.1, pp.101-138, 2009.
DOI : 10.1016/j.anihpc.2007.07.004

URL : https://hal.archives-ouvertes.fr/hal-00150455

R. Hempel, L. Seco, and B. Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, Journal of Functional Analysis, vol.102, issue.2, pp.448-483, 1991.
DOI : 10.1016/0022-1236(91)90130-W

M. Hoffmann-ostenhof, T. Hoffmann-ostenhof, and N. Nadirashvili, The nodal line of the second eigenfunction of the Laplacian in R 2 2 can be closed, Duke Math, J, vol.90, issue.3, pp.631-640, 1997.

D. Holcman, A. Marchewka, and Z. Schuss, Survival probability of diffusion with trapping in cellular neurobiology, Physical Review E, vol.72, issue.3, p.31910, 2005.
DOI : 10.1103/PhysRevE.72.031910

A. S. Householder, Unitary Triangularization of a Nonsymmetric Matrix, Journal of the ACM, vol.5, issue.4, pp.339-342, 1958.
DOI : 10.1145/320941.320947

URL : https://hal.archives-ouvertes.fr/hal-01316095

B. D. Hughes, Random Walks and Random Environments, 1995.

D. Jakobson, N. Nadirashvili, and J. Toth, Geometric properties of eigenfunctions, Russian Mathematical Surveys, vol.56, issue.6, pp.1085-1105, 2001.
DOI : 10.1070/RM2001v056n06ABEH000453

D. Jakobson and N. Nadirashvili, Eigenfunctions with few critical points, Journal of Differential Geometry, vol.53, issue.1, pp.177-182, 1999.
DOI : 10.4310/jdg/1214425450

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.5078

D. Jerison, The first nodal line of a convex planar domain, Duke Math, J. Internat. Math. Res. Notices, vol.62, issue.1, pp.1-5, 1991.

S. Jimbo, The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary condition, Journal of Differential Equations, vol.77, issue.2, pp.322-350, 1989.
DOI : 10.1016/0022-0396(89)90147-2

S. Jimbo and Y. Morita, Remarks on the behavior of certain eigenvalues on a singularly pertubed domain with several thin channels, Comm. Part. Diff. Eq, vol.17, pp.523-552, 1992.

S. Jimbo, Perturbation formula of eigenvalues in a singularly perturbed domain, Journal of the Mathematical Society of Japan, vol.45, issue.2, pp.339-356, 1993.
DOI : 10.2969/jmsj/04520339

D. S. Jones, The eigenvalues of 2 u + ?? u=0 when the boundary conditions are given on semi-infinite domains, Mathematical Proceedings of the Cambridge Philosophical Society, vol.47, issue.04, pp.668-684, 1953.
DOI : 10.1007/BF01447273

A. R. Kansal and S. Torquato, Prediction of trapping rates in mixtures of partially absorbing spheres, The Journal of Chemical Physics, vol.116, issue.24, p.10589, 2002.
DOI : 10.1063/1.1479718

L. Kaplan and E. Heller, Weak quantum ergodicity, Phys, Proc. Nat. Ac, pp.1-18, 1998.
DOI : 10.1016/s0167-2789(98)00156-0

URL : http://arxiv.org/abs/chao-dyn/9810002

R. F. Kayser and J. B. Hubbard, Diffusion in a Medium with a Random Distribution of Static Traps, Physical Review Letters, vol.51, issue.2, p.79, 1983.
DOI : 10.1103/PhysRevLett.51.79

R. F. Kayser and J. B. Hubbard, Reaction diffusion in a medium containing a random distribution of nonoverlapping traps, The Journal of Chemical Physics, vol.80, issue.3, p.1127, 1984.
DOI : 10.1063/1.446841

J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Annals of Physics, vol.9, issue.1, pp.24-75, 1960.
DOI : 10.1016/0003-4916(60)90061-0

E. T. Kippatrick, Tables of values of the modified Mathieu functions, Mathematics of Computation, vol.14, issue.70, pp.118-129, 1960.
DOI : 10.1090/S0025-5718-1960-0113288-4

T. Kolokolnikov, M. S. Titcombe, and M. J. Ward, Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps, European Journal of Applied Mathematics, vol.16, issue.2, pp.16-161, 2005.
DOI : 10.1017/S0956792505006145

P. Kröger, On the ground state eigenfunction of a convex domain in Euclidean space, Potential Analysis, vol.294, issue.1, pp.103-108, 1996.
DOI : 10.1007/BF00276699

M. Lapidus and F. Drum, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Transactions of the American Mathematical Society, vol.325, issue.2, pp.325-465, 1991.
DOI : 10.1090/S0002-9947-1991-0994168-5

M. L. Lapidus and M. M. Pang, Eigenfunctions of the Koch snowflake domain, Communications in Mathematical Physics, vol.47, issue.no. 5, pp.359-376, 1995.
DOI : 10.1007/BF02099432

M. L. Lapidus, J. W. Neuberger, R. J. Renka, and C. A. Griffith, SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS, International Journal of Bifurcation and Chaos, vol.06, issue.07, pp.1185-1210, 1996.
DOI : 10.1142/S0218127496000680

V. F. Lazutkin, Construction of an asymptotic series of eigenfunctions of the bouncing ball type, Proc. Steklov Inst, pp.106-118, 1968.

V. F. Lazutkin, THE EXISTENCE OF CAUSTICS FOR A BILLIARD PROBLEM IN A CONVEX DOMAIN, Mathematics of the USSR-Izvestiya, vol.7, issue.1, pp.185-214, 1973.
DOI : 10.1070/IM1973v007n01ABEH001932

V. F. Lazutkin, Vypuklyj billiard i sobstvennye funkcii operatora Laplasa (Leningr. univ., 1981)

V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Ergebnisse der Mathematik und ihrer Grenzgebiete, 1993.
DOI : 10.1007/978-3-642-76247-5

S. B. Lee, I. C. Kim, C. A. Miller, and T. S. , Random-walk simulation of diffusion-controlled processes among static traps, Physical Review B, vol.39, issue.16, p.11833, 1989.
DOI : 10.1103/PhysRevB.39.11833

W. R. Leeb, Algorithm 537: Characteristic Values of Mathieu's Differential Equation [S22], ACM Transactions on Mathematical Software, vol.5, issue.1, pp.112-117, 1979.
DOI : 10.1145/355815.355824

P. E. Levitz, D. S. Grebenkov, M. Zinsmeister, K. Kolwankar, and B. Sapoval, Brownian Flights over a Fractal Nest and First-Passage Statistics on Irregular Surfaces, Physical Review Letters, vol.96, issue.18, p.180601, 2006.
DOI : 10.1103/PhysRevLett.96.180601

URL : https://hal.archives-ouvertes.fr/hal-00079193

C. S. Lin, On the second eigenfunctions of the Laplacian in R2, Communications in Mathematical Physics, vol.24, issue.2, pp.161-166, 1987.
DOI : 10.1007/BF01217758

C. M. Linton and P. Mciver, Embedded trapped modes in water waves and acoustics, Wave Motion, vol.45, issue.1-2, pp.16-29, 2007.
DOI : 10.1016/j.wavemoti.2007.04.009

J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, 1972.
DOI : 10.1007/978-3-642-65161-8

C. C. Liu, T. H. Lu, Y. F. Chen, and K. F. Huang, Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards, Physical Review E, vol.74, issue.4, p.46214, 2006.
DOI : 10.1103/PhysRevE.74.046214

S. N. Majumdar, Brownian Functionals in Physics and Computer Science, Curr. Sci, vol.89, p.2076, 2005.
DOI : 10.1142/9789812772718_0006

URL : https://hal.archives-ouvertes.fr/hal-00165789

J. Marlof and Z. Rudnick, Almost all eigenfunctions of a rational polygon are uniformly distributed Journal of, Spectral Theory, vol.2, pp.107-113, 2012.

V. P. Maslov, Perturbation theory and asymptotic methods, 1965.

V. P. Maslov, The behavior at infinity of eigenfunctions of the Schrödinger equation

P. N. Mcgraw and M. Menzinger, Laplacian spectra as a diagnostic tool for network structure and dynamics, Physical Review E, vol.77, issue.3, p.31102, 2008.
DOI : 10.1103/PhysRevE.77.031102

A. D. Melas, On the nodal line of the second eigenfunction of the Laplacian in $\mathbf{R}^2$, Journal of Differential Geometry, vol.35, issue.1, pp.255-263, 1992.
DOI : 10.4310/jdg/1214447811

C. A. Miller and S. Torquato, Diffusion-controlled reactions among spherical traps: Effect of polydispersity in trap size, Physical Review B, vol.40, issue.10, p.7101, 1989.
DOI : 10.1103/PhysRevB.40.7101

C. A. Miller, I. C. Kim, and S. Torquato, Trapping and flow among random arrays of oriented spheroidal inclusions, The Journal of Chemical Physics, vol.94, issue.8, p.5592, 1991.
DOI : 10.1063/1.460495

G. M. Armando and . Neves, Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations, Communications on Pure and Applied Analysis, vol.9, issue.3, pp.611-624, 2010.

B. T. Nguyen and D. S. Grebenkov, Localization of Laplace eigenfunctions in circular, spherical and elliptical domains

B. T. Nguyen, L. T. Duc, N. D. Thuc, and B. V. Thach, A divide-and-conquer algorithm for a symmetric tri-block-diagonal matrix, 2012 Proceedings of IEEE Southeastcon, p.2012, 2012.
DOI : 10.1109/SECon.2012.6196898

B. T. Nguyen and D. S. Grebenkov, A Spectral Approach to Survival Probabilities in??Porous??Media, Journal of Statistical Physics, vol.68, issue.3, pp.532-554, 2010.
DOI : 10.1007/s10955-010-0054-1

O. Olendski and L. Mikhailovska, Theory of a curved planar waveguide with Robin boundary conditions, Physical Review E, vol.81, issue.3, p.36606, 2010.
DOI : 10.1103/PhysRevE.81.036606

F. W. Olver, A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order, Proc. Cambridge Philos. Soc, pp.47-699, 1951.
DOI : 10.1002/asna.18911282404

F. W. Olver, Some new asymptotic expansions for Bessel functions of large orders, Proc. Cambridge Philos, pp.414-427, 1952.
DOI : 10.1103/PhysRev.80.1112

Y. B. Orocko, ON THE APPLICATION OF SPECTRAL THEORY TO OBTAIN ESTIMATES OF SOLUTIONS OF THE SCHR??DINGER EQUATION, Mathematics of the USSR-Sbornik, vol.22, issue.2, pp.167-186, 1974.
DOI : 10.1070/SM1974v022n02ABEH001690

M. Pang, Approximation of ground state eigenfunction on the snowflake region , Bull. London, Math. Soc, vol.28, pp.488-494, 1996.

M. Pang, Approximation of ground state eigenvalues and eigenfunctions of Dirichlet Laplacians, Bull. London, Math. Soc, vol.29, pp.720-730, 1997.

R. Parker, Resonance effects in wake shedding from parallel plates: Some experimental observations, Journal of Sound and Vibration, vol.4, issue.1, pp.62-72, 1966.
DOI : 10.1016/0022-460X(66)90154-4

R. Parker, Resonance effects in wake shedding from parallel plates: Calculation of resonant frequencies, Journal of Sound and Vibration, vol.5, issue.2, pp.330-343, 1967.
DOI : 10.1016/0022-460X(67)90113-7

L. Payne, On two conjectures in the fixed membrane eigenvalue problem, ZAMP Zeitschrift f??r angewandte Mathematik und Physik, vol.39, issue.5, pp.720-729, 1973.
DOI : 10.1007/BF01597076

S. Pillay, M. J. Ward, A. Peirce, and T. Kolokolnikov, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains, Multiscale Modeling & Simulation, vol.8, issue.3, pp.803-835, 2010.
DOI : 10.1137/090752511

M. A. Pinsky, The Eigenvalues of an Equilateral Triangle, SIAM Journal on Mathematical Analysis, vol.11, issue.5, pp.819-827, 1980.
DOI : 10.1137/0511073

M. A. Pinsky, Completeness of the Eigenfunctions of the Equilateral Triangle, SIAM Journal on Mathematical Analysis, vol.16, issue.4, pp.848-851, 1985.
DOI : 10.1137/0516063

T. Prosen and M. Robnik, Survey of the eigenfunctions of a billiard system between integrability and chaos, Journal of Physics A: Mathematical and General, vol.26, issue.20, p.5365, 1993.
DOI : 10.1088/0305-4470/26/20/021

R. Putter, On the nodal lines of second eigenfunctions of the fixed membrane problem, Commentarii Mathematici Helvetici, vol.65, issue.1, pp.96-103, 1990.
DOI : 10.1007/BF02566596

C. V. Raman and G. A. Sutherland, Whispering-Gallery Phenomena at St. Paul's Cathedral, Nature, vol.108, issue.2706, p.42, 1921.
DOI : 10.1038/108042a0

C. V. Raman and G. A. Sutherland, On the Whispering-Gallery Phenomenon, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.100, issue.705, pp.424-428, 1922.
DOI : 10.1098/rspa.1922.0007

S. Redner, A Guide to First-Passage Processes, 2001.

P. M. Richards, Diffusion to Finite-Size Traps, Physical Review Letters, vol.56, issue.17, p.1838, 1986.
DOI : 10.1103/PhysRevLett.56.1838

P. M. Richards, Diffusion to nonoverlapping or spatially correlated traps, Physical Review B, vol.35, issue.1, p.248, 1987.
DOI : 10.1103/PhysRevB.35.248

P. M. Richards and S. Torquato, Upper and lower bounds for the rate of diffusion???controlled reactions, The Journal of Chemical Physics, vol.87, issue.8, p.4612, 1987.
DOI : 10.1063/1.452872

J. Rubinstein and S. Torquato, Diffusion???controlled reactions: Mathematical formulation, variational principles, and rigorous bounds, The Journal of Chemical Physics, vol.88, issue.10, p.6372, 1988.
DOI : 10.1063/1.454474

Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Communications in Mathematical Physics, vol.55, issue.1, pp.195-213, 1994.
DOI : 10.1007/BF02099418

S. Russ and B. Sapoval, Increased damping of irregular resonators, Physical Review E, vol.65, issue.3, p.36614, 2002.
DOI : 10.1103/PhysRevE.65.036614

S. Russ, B. Sapoval, and O. Haeberle, Irregular and fractal resonators with Neumann boundary conditions: Density of states and localization, Physical Review E, vol.55, issue.2, p.1413, 1997.
DOI : 10.1103/PhysRevE.55.1413

N. Saito, Data analysis and representation on a general domain using eigenfunctions of Laplacian, Applied and Computational Harmonic Analysis, vol.25, issue.1, pp.68-97, 2008.
DOI : 10.1016/j.acha.2007.09.005

B. Sapoval, Experimental observation of local modes in fractal drums, Physica D: Nonlinear Phenomena, vol.38, issue.1-3, pp.296-298, 1989.
DOI : 10.1016/0167-2789(89)90209-1

B. Sapoval, T. Gobron, and A. Margolina, Vibrations of fractal drums, Physical Review Letters, vol.67, issue.21, p.2974, 1991.
DOI : 10.1103/PhysRevLett.67.2974

B. Sapoval and T. Gobron, Vibrations of strongly irregular or fractal resonators, Physical Review E, vol.47, issue.5, p.3013, 1993.
DOI : 10.1103/PhysRevE.47.3013

B. Sapoval, O. Haeberle, and S. Russ, Acoustical properties of irregular and fractal cavities, The Journal of the Acoustical Society of America, vol.102, issue.4
DOI : 10.1121/1.419653

E. E. Schnol, On Behaviour of the eigenfunctions of Schrödinger's Equation, Mat. Sb, vol.4284, pp.273-286, 1957.

R. L. Schult, D. G. Ravenhall, and H. W. Wyld, Quantum bound states in a classically unbound system of crossed wires, Physical Review B, vol.39, issue.8, pp.5476-5479, 1989.
DOI : 10.1103/PhysRevB.39.5476

J. Schwinger, ON THE BOUND STATES OF A GIVEN POTENTIAL, Proceedings of the National Academy of Sciences, vol.47, issue.1, pp.122-129, 1961.
DOI : 10.1073/pnas.47.1.122

R. B. Shirts, The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order, ACM Transactions on Mathematical Software, vol.19, issue.3, pp.377-390, 1993.
DOI : 10.1145/155743.155796

A. Shnirelman, Ergodic properties of eigenfunctions, Uspechi Math, Nauk, vol.29, pp.181-182, 1974.

K. M. Siegel, An inequality involving Bessel functions of argument nearly equal to their order, Proceedings of the American Mathematical Society, vol.4, issue.6
DOI : 10.1090/S0002-9939-1953-0058775-0

Y. G. Sinai, Dynamical systems with elastic reflections, Russian Mathematical Surveys, vol.25, issue.2, pp.137-189, 1970.
DOI : 10.1070/RM1970v025n02ABEH003794

A. Singer, Z. Schuss, D. Holcman, R. S. Eisenberg, and N. Escape, Narrow Escape, Part I, Journal of Statistical Physics, vol.102, issue.12, p.437, 2006.
DOI : 10.1007/s10955-005-8026-6

A. Singer, Z. Schuss, D. Holcman, and N. Escape, Part II, J. Stat. Phys, vol.122, p.465, 2006.
DOI : 10.9783/9781512806960-003

URL : https://hal.archives-ouvertes.fr/hal-00318200

J. Slemons, Toward the Solution of the Eigenproblem: Nonsymmetric Tridiagonal Matrices PhD thesis, 2007.

S. Sridhar, Experimental observation of scarred eigenfunctions of chaotic microwave cavities, Physical Review Letters, vol.67, issue.7, pp.785-788, 1991.
DOI : 10.1103/PhysRevLett.67.785

S. Sridhar, D. O. Hogenboom, and B. A. Willemsen, Microwave experiments on chaotic billiards, Journal of Statistical Physics, vol.65, issue.1-2, pp.239-258, 1992.
DOI : 10.1007/BF01048844

G. Timp, H. U. Baranger, P. Devegvar, J. E. Cunningham, R. E. Howard et al., Propagation around a Bend in a Multichannel Electron Waveguide, Physical Review Letters, vol.60, issue.20, pp.2081-2084, 1988.
DOI : 10.1103/PhysRevLett.60.2081

S. Torquato and I. C. Kim, Efficient simulation technique to compute effective properties of heterogeneous media, Applied Physics Letters, vol.55, issue.18, p.1847, 1989.
DOI : 10.1063/1.102184

S. Torquato and M. Avellaneda, Diffusion and reaction in heterogeneous media: Pore size distribution, relaxation times, and mean survival time, The Journal of Chemical Physics, vol.95, issue.9, p.6477, 1991.
DOI : 10.1063/1.461519

S. Torquato, Diffusion and reaction among traps: some theoretical and simulation results, Journal of Statistical Physics, vol.28, issue.5-6, p.1173, 1991.
DOI : 10.1007/BF01049606

S. Torquato and C. L. Yeong, Universal scaling for diffusion-controlled reactions among traps, The Journal of Chemical Physics, vol.106, issue.21, p.8814, 1997.
DOI : 10.1063/1.473941

J. Toth and S. Zelditch, Counting nodal lines which touch the boundary of an analytic domain, Journal of Differential Geometry, vol.81, issue.3, pp.649-686, 2009.
DOI : 10.4310/jdg/1236604347

A. Truman and D. Williams, Diffusion Processes and Related Problems in Analysis, 1990.

G. Veble, M. Robnik, and J. Liu, Study of regular and irregular states in generic systems, Journal of Physics A: Mathematical and General, vol.32, issue.36, pp.6423-6444, 1999.
DOI : 10.1088/0305-4470/32/36/306

M. J. Ward, Asymptotic Methods for Reaction-Diffusion Systems: Past and Present, Bulletin of Mathematical Biology, vol.68, issue.3, p.1151, 2006.
DOI : 10.1007/s11538-006-9091-y

G. N. Watson, A treatise on the theory of Bessel functions (Cambridge Mathematical Library, 1995.

G. H. Weiss, Aspects and Applications of the Random Walk, 1994.

G. H. Weiss, Overview of theoretical models for reaction rates, Journal of Statistical Physics, vol.14, issue.1-2, 1986.
DOI : 10.1007/BF01010838

J. Wiersig, Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles, Physical Review A, vol.84, issue.6, p.63828, 2011.
DOI : 10.1103/PhysRevA.84.063828

S. T. Yau, Problem Section, Seminar on Differential Geometry, Anna. of Math. Stud, vol.102, pp.669-706, 1982.

S. T. Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds, Proc. Symp. Pure Math, pp.1-28, 1993.

S. T. Yau, A note on the distribution of critical points of eigenfunctions, Tsing Hua Lectures in Geometry and Analysis, pp.315-317, 1997.

S. B. Yuste, G. Oshanin, K. Lindenberg, O. Bénichou, and J. Klafter, Survival probability of a particle in a sea of mobile traps: A tale of tails, Physical Review E, vol.78, issue.2, p.21105, 2008.
DOI : 10.1103/PhysRevE.78.021105

S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Communications in Mathematical Physics, vol.55, issue.6, pp.673-682, 1996.
DOI : 10.1007/BF02099513

S. Zelditch, Quantum Mixing, Journal of Functional Analysis, vol.140, issue.1, pp.68-86, 1996.
DOI : 10.1006/jfan.1996.0098

R. Zwanzig and A. Szabo, Time dependent rate of diffusion-influenced ligand binding to receptors on cell surfaces, Biophysical Journal, vol.60, issue.3, pp.671-678, 1991.
DOI : 10.1016/S0006-3495(91)82096-3