Initial length scale estimate for the Schrödginer operator with a random fast oscillating potential in a multi-dimensional layer

  • D. I. Borisov University of Hradec Králové, Hradec Králové, Czech Republic


We consider the Dirichlet Laplacian in a multi-dimensional layer located between two parallel hyperplanes of codimension one. Such operator is perturbed by a fast oscillating random potential. Namely, the layer is partitioned into periodicity cells by a given periodic lattice and in each cell we consider a fast oscillating potential depending on a random variable multiplied by a global small parameters. All random variables associated with the periodicity cells are assumed to be independent and identically distributed. The fast oscillating potential introduced in the way standard for the homogenization theory. Namely, it depends on slow and fast variables, is compactly supported w.r.t. the slow variables and is periodic w.r.t. the fast ones. The main obtained result is the initial length scale estimate for the considered operator. Such estimate is the induction base for proving the spectral localization at the bottom of the spectrum by the multiscale analysis.


[1] D.I. Borisov «Initial length scale estimate for layers with small random negative definite perturbations», J. Math. Sci. (2017 to appear).
[2] D.I. Borisov, R.Kh. Karimov, T.F. Sharapov «Initial length scale estimate for waveguides with some random singular potentials», Ufa Math. J. Vol.7, No.2.(2015) : 33–54.
[3] N.S. Bakhvalov, G.P. Panasenko Homogenization: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials (Nauka, Moscow, 1984). Translated by Kluwer (Dordrecht, 1989).
[4] F. Martinelli and H. Holden «On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L 2 (R ν )», Comm. Math. Phys. Vol. 93, No. 2.(1984): 197-217.
[5] J. Fröhlich, Т. Spencer «Absence of diffusion in the Anderson tight binding model for large disorder or low energy», Comm. Math. Phys. Vol. 88, No. 2. (1983) : 151-184.
[6] J. Baker, M. Loss, and G. Stolz «Minimizing the ground state energy of an electron in a randomly deformed lattice», Comm. Math. Phys. Vol. 283, No. 2. (2008) : 397–415.
[7] D. Borisov and I. Veselić «Low lying spectrum of weak-disorder quantum waveguides», J. Stat. Phys. Vol. 142, No. 1. (2011) : 58–77.
[8] D. Borisov and I. Veselić «Low lying eigenvalues of randomly curved quantum waveguides», J. Funct. Anal. Vol. 265, No. 11. (2013) : 2877–2909.
[9] J. Bourgain «An approach to Wegner’s estimate using subharmonicity», J. Stat. Phys. Vol. 134, No. 5-6. (2009) : 969–978.
[10] L. Erdös and D. Hasler «Anderson localization at band edges for random magnetic fields», J. Stat. Phys. Vol. 146, No. 5. (2012) : 900–923.
[11] L. Erdös and D. Hasler «Wegner estimate and anderson localization for random magnetic fields», Comm. Math. Phys. Vol. 309, No. 2. (2012) : 507–542.
[12] F. Ghribi, P. D. Hislop, and F. Klopp «Localization for Schrödinger operators with random vector potentials», In Adventures in mathematical physics. Contemp. Math. Providence, RI, Amer. Math. Soc. Vol. 447. (2007) : 123–138.
[13] F. Ghribi and F. Klopp «Localization for the random displacement model at weak disorder», Ann. Henri Poincaré Vol. 11, No. 1-2. (2010) : 127–149.
[14] P.D. Hislop and F.Klopp «The integrated density of states for some random operators with nonsign definite potentials», J. Funct. Anal. Vol. 195, No. 1. (2002) : 12–47.
[15] F. Kleespies and P. Stollmann «Lifshitz asymptotics and localization for random quantum waveguides», Rev. Math. Phys. Vol. 12, No. 10. (2000) : 1345–1365.
[16] F. Klopp «Localization for semiclassical continuous random Schrödinger operators II, No. The random displacement model», Helv. Phys. Acta. Vol. 66, No. 7-8. (1993) : 810–841.
[17] F. Klopp «Localisation pour des opérateurs de Schrödinger aléatoires dans L 2 (R d ): un modéle semi-classique», Ann. Inst. Fourier (Grenoble) Vol. 45, No. 1. (1995) : 265–316.
[18] F. Klopp «Localization for some continuous random Schrödinger operators», Comm. Math. Phys. Vol. 167, No. 3. (1995) : 553–569.
[19] F. Klopp «Weak disorder localization and Lifshitz tails: continuous Hamiltonians», Ann. Henri Poincaré Vol. 3, No. 4. (2002) : 711–737.
[20] F. Klopp, M. Loss, S. Nakamura, and G. Stolz «Localization for the random displacement model», Duke Math. J. Vol. 161, No. 4. (2012) : 587–621.
[21] F. Klopp and S. Nakamura «Spectral extrema and Lifshitz tails for non-monotonous alloy type models», Comm. Math. Phys. Vol. 287, No. 3. (2009) : 1133–1143.
[22] F. Klopp, S. Nakamura, F. Nakano, and Y. Nomura «Anderson localization for 2D discrete Schrödinger operators with random magnetic fields», Ann. Henri Poincaré Vol. 4, No. 4. (2003) : 795–811.
[23] V. Kostrykin and I. Veselić «On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials», Math. Zeit. Vol. 252, No. 2. (2006) : 367–392.
[24] D. Lenz, N. Peyerimhoff, O. Post, and I. Veselić «Continuity properties of the integrated density of states on manifolds», Japan. J. Math. Vol. 3, No. 1. (2008) : 121–161.
[25] D. Lenz, N. Peyerimhoff, O. Post, and I. Veselić «Continuity of the integrated density of states on random length metric graphs», Math. Phys. Anal. Geom. Vol. 12, No. 3 (2009) : 219-254.
[26] D. Lenz, N. Peyerimhoff, and I. Veselić «Integrated density of states for random metrics on manifolds», Proc. London Math. Soc. Vol. 88, No. 3. (2004) : 733–752.
[27] K. Leonhardt, N. Peyerimhoff, M. Tautenhahn, and I. Veselić «Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials», Rev. Math. Phys. Vol. 27, No. 4. (2015) : 1550007.
[28] G. Stolz «Non-monotonic random Schrödinger operators: the Anderson model», J. Math. Anal. Appl. Vol. 248, No. 1. (2000) : 173–183.
[29] N. Ueki «On spectra of random Schrödinger operators with magnetic fields», Osaka J. Math. Vol. 31, No. 1. (1994) : 177–187.
[30] N. Ueki «Simple examples of Lifschitz tails in Gaussian random magnetic fields», Ann. Henri Poincaré Vol. 1, No. 3. (2000) : 473–498.
[31] N. Ueki «Wegner estimate and localization for random magnetic fields», Osaka J. Math. Vol. 45, No. 3. (2008) : 565–608.
[32] I. Veselić «Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators», Lett. Math. Phys. Vol. 45, No. 3. (2002) : 199–214.
[33] D. Borisov, A. Golovina, I. Veselić «Quantum Hamiltonians with weak random abstract perturbation. I. Initial length scale estimate», Ann. H. Poincaré Vol. 17, No. 9. (2016) : 2341–2377.
How to Cite
BORISOV, D. I.. Initial length scale estimate for the Schrödginer operator with a random fast oscillating potential in a multi-dimensional layer. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 93, n. 1, p. 21-31, june 2018. ISSN 2617-4871. Available at: <>. Date accessed: 20 jan. 2021.
Keywords random Hamiltonian, fast oscillating potential, initial length scale estimate, small parameter, multi-dimensional layer