Band edge localization and the density of states for acoustic and electromagnetic waves in random media

 Access to full text

 Full (PDF)

1. [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations, Mathematical Notes 29, Princeton University Press, Princeton, NJ, 1982. Zbl0503.35001MR745286
2. [2] M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys., Vol. 6, N° 5a, 1994, pp. 1163-1182. Zbl0843.47039MR1301371
3. [3] J.-M. Barbaroux, thèse, Université de Toulon et du Var1997. 
4. [4] J.-M. Barbaroux, J.M. Combes and P.D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta, Vol. 70, 1997, pp. 16-43. Zbl0866.35077MR1441595
5. [5] J.-M. Barbaroux, J.M. Combes and P.D. Hislop, Landau Hamiltonians with unbounded random potentials, Lett. Math. Phys., Vol. 40, 1997. Zbl0894.47028MR1453246
6. [6] D. Belitz and T.R. Kirkpatrick, the Anderson-Mott Transition, Rev. of Modern Phys., Vol. 66, 1994, p. 261. 
7. [7] M.S. Birman and M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Dordrecht: D. Reidel Publishing Co, 1987. MR345774
8. [8] J.-M. Combes and P.D. Hislop, Localization for some continuous random Hamiltonians in d-dimensions, J. Funct. Anal., Vol. 124, 1994, pp. 149-180. Zbl0801.60054MR1284608
9. [9] J.-M. Combes and P.D. Hislop, Localization Properties of Continuous Disordered Systems in d-dimensions, Proc. of "Mathematical Quantum Theory", Vancouver, Canada, 1993. Zbl0820.60089MR1332042
10. [10] J.-M. Combes, P.D. Hislop and E. Mourre, Spectral Averaging, Perturbation of Singular Spectra, and Localization, Trans. Amer. Math. Soc., Vol. 348, 1996, pp. 4883-4894. Zbl0868.35081MR1344205
11. [11] J.-M. Combes, P.D. Hislop and E. Mourre, Correlated Wegner Inequalities for Long-range and Correlated Potentials, preprint 1997. 1997. 
12. [12] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990. Zbl0717.60074MR1102675
13. [13] J.-M. Combes and L. Thomas, Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators, Commun. Math. Phys., Vol. 34, 1973, pp. 251-276. Zbl0271.35062MR391792
14. [14] P.A. Deift and R. Hempel, On the existence of eigenvalues of the Schrödinger operator H + λW in a gap of σ(H), Commun. Math. Phys., Vol. 103, 1986, pp. 461-490. Zbl0594.34022MR832922
15. [15] H.J.S. Dorren and A. Tip, Maxwell's equations for non-smooth media; fractal-shaped and pointlike objects, J. Math. Phys., Vol. 32, 1991, p. 3060. MR1131689
16. [16] W. Faris, Localization for a random discrete wave equation, in Random Media, IMA Volume 7, G. Papanicolaou, ed., Springer-Verlag, New York, 1987. Zbl0656.35102MR901043
17. [17] W. Faris, A localization principle for multiplicative perturbations, J. Funct. Anal., Vol. 67, 1986, pp. 105-114. Zbl0588.47016MR842605
18. [18] A. Figotin and A. Klein, Localization phenomenon in gaps of the spectrum of random lattice operators, J. Stat. Phys., Vol. 75, 1994, pp. 997-1021. Zbl0838.47050MR1285295
19. [19] A. Figotin and A. Klein, Localization of Electromagnetic and Acoustic Waves in Random Media: Lattice Models, J. Stat. Phys., Vol. 76, 1994, pp. 985-1003. Zbl0839.76077MR1299235
20. [20] A. Figotin and A. Klein, Localization of Classical Waves I: Acoustic Waves, Commun. Math. Phys., Vol. 180, 1996, pp. 439-482. Zbl0878.35109MR1405959
21. [21] A. Figotin and A. Klein, Localization of Classical Waves II: Electromagnetic Waves, Commun. Math. Phys., Vol. 184, 1997, pp. 411-441. Zbl0878.35110MR1462752
22. [22] A. Figotin and P. Kuchment, Band-Gap structure of spectra of periodic dielectric and acoustic media. I. scalar Model, SIAM J. Appl. Math., Vol. 56, No. 1, 1996, pp. 68-88. Zbl0852.35014MR1372891
23. [23] A. Figotin and P. Kuchment, Band-Gap structure of spectra of periodic dielectric and acoustic media. II. 2D Photonic Crystals, SIAM J. Appl. Math., Vol. 56, No. 6, 1996, pp. 1561-1620. Zbl0868.35009MR1417473
24. [24] J. Frölich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys., Vol. 88, 1983, pp. 151-184. Zbl0519.60066MR696803
25. [25] F. Gesztesy, Some applications of commutation methods, in Schrödinger operators: Proc. of the Nordic Summer School in Mathematics held at Sandbjerg Slot, Sonderborg, Denmark, 1988, Lecture Notes in Physics No. 345, H. Holden and A. Jensen, eds., Berlin: Springer-Verlag1989. Zbl0716.35075MR1037318
26. [26] J. Howland, Perturbation theory of dense point spectra, J. Funct. Anal., Vol. 74, 1987, pp. 52-80. Zbl0646.47011MR901230
27. [27] S. John, The localization of light and other classical waves in disordered media, Comments Cond. Mat. Phys., Vol. 14, 1988, pp. 193-230. 
28. [28] W. Kirsch and F. Martinelli, On the spectrum of Schrödinger operators with a random potential, Commun. Math. Phys., Vol. 85, 1982, pp. 329-350. Zbl0506.60058MR678150
29. [29] W. Kirsch, P. Stollnian and G. Stoltz, Localization for random perturbations ofperiodic Schrödinger operators, preprint 1996. MR1381595
30. [30] F. Klopp, Internal Lifshits tails for random perturbations of periodic Schrödinger operators, preprint 1997. Zbl1060.82509MR1695202
31. [31] S. Kotani and B. Simon, Localization in general one-dimensional systems. II, Commun. Math. Phys., Vol. 112, 1987, pp. 103-120. Zbl0637.60080MR904140
32. [32] P. Kuchment, Floquet Theory for Partial Differential Equations, Basel: Birkhäuser Verlag1993. Zbl0789.35002MR1232660
33. [33] F.H. Lin, A uniqueness equation for parabolic equations, Comm. Pure Appl. Math., Vol. 43, 1990, pp. 127-136. Zbl0727.35063
34. [34] L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Berlin: Springer-Verlag1992. Zbl0752.47002MR1223779
35. [35] B. Simon, Trace ideals and their applications, Cambridge: Cambridge University Press1979. Zbl0423.47001MR541149
36. [36] B. Sinion and T. Wolff, Singular continuous spectrum under rank one perturbation and localization for random Hamiltonians, Comm. Pure Appl. Math., Vol. 39, 1986, pp. 75-90. Zbl0609.47001MR820340
37. [37] T. Spencer, Localization for random and quasi-periodic potentials, J. Stat. Phys., Vol. 51, 1988, pp. 1009-1019. Zbl1086.82547MR971043
38. [38] P. Stollmann, Localization for random perturbations of anisotropic periodic media, to appear in Israel Journal of Mathematics1997. Zbl0918.35042MR1658543
39. [39] D.V. Van Coevorden, R. Sprik, A. Tip and A. Lagendijk, Photonic Bandstructure of Atomic Lattices, Phys. Rev. Letters, Vol. 77, 1996, p. 2412. 
40. [40] H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys., Vol. 124, 1989, pp. 285-299. Zbl0698.60051MR1012868