Semi-classical Asymptotics for the Schrödinger Operator with Oscillating Decaying Potential

Dimassi, Mouez

Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 734-747

Résumé

We study the distribution of the discrete spectrumof the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$ .

Détail

DOI : 10.4153/CMB-2016-022-8

Mots-clés : 81Q10, 35P20, 47A55, 47N50, 81Q15, periodic Schrödinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift function

Bibliographie

[1] [1] Birman, M. Sh. and Solomyak, M., On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential. J. Anal. Math. 83(2001), 337–391. Google Scholar | DOI

[2] [2] Birman, M. Sh., Laptev, A., and Suslina, T. A., Discrete spectrum of the two-dimensional periodic elliptic second order operator perturbed by a decreasing potential. I. Semi-infinite gap. St. Petersburg Math. J. 12(2001), 535–567. Google Scholar

[3] [3] Borisov, D. I., The spectrum of the Schrodinger operator perturbed by a rapidly oscillating potential. J. Math. Sci. (N. Y.) 139(2006), no. 1, 6243–6323. http://dx.doi.Org/10.1007/s10958-006-0349-6 Google Scholar

[4] [4] Borisov, D. I. and Gadyl'shin, R. R., On the spectrum of the Schrodinger operator with a rapidly oscillating compactly supported potential. (Russian) Teoret. Mat. Fiz. 147(2006), no. 1, 58–63; translation in Theoret. and Math. Phys. 147(2006), no. 1, 496-500. http://dx.doi.Org/10.4213/tmf2O22 Google Scholar

[5] [5] Borisov, D. I. and Gadyl'shin, R. R., On the spectrum of a selfadjoint differential operator with rapidly oscillating coefficients on the axis. (Russian) Mat. Sb. 198 (2007), no. 8, 3–34; translation in Sb. Math. 198 (2007), no. 7-8, 1063-1093 Google Scholar | DOI

[6] [6] Buslaev, V. S., Semiclassical approximation for equations with periodic coefficients. (Russian) Math. Surveys 42 (1987), no. 6, 97–125. Google Scholar

[7] [7] Christ, M. and Kiselev, A., Absolutely continuous spectrum for one-dimensional Schrodinger operators with slowly decaying potentials: some optimal results. J. Amer. Math. Soc. 11(1998), no. 4, 771–797. Google Scholar | DOI

[8] [8] Devinatz, A. and Rejto, P. A limiting absorption principle for Schrodinger operators with oscillating potentials, Parti. J. Differential Equations 49(1983), no. 1, 29–84. http://dx.doi.Org/10.1016/0022-0396(83)90019-0 Google Scholar

[9] [9] Denisov, S., Absolutely continuous spectrum of multidimensional Schrodinger operator. Int. Math. Res. Not. (2004), no. 74, 3963–3982. http://dx.doi.Org/10.1155/S107379280414141X Google Scholar

[10] [10] Dimassi, M., Developpements asymptotiques des perturbations lentes de Voperateur de Schrodinger periodique. Comm. Partial Differential Equations 18(1993), no. 5-6, 771–803. http://dx.doi.Org/10.1080/03605309308820950 Google Scholar

[11] [11] Dimassi, M., Trace asymptotics formulas and some applications. Asymptot. Anal. 18(1998), no. 1-2, 1–32. Google Scholar

[12] [12] Dimassi, M., Resonances for a slowly varying perturbation of a periodic Schrodinger operator. Canad. J. Math. 54(2002), no. 5, 998–1037. Google Scholar | DOI

[13] [13] Dimassi, M. and Sjostrand, J., Spectral asymptotics in the semi-classical limit. London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999. Google Scholar

[14] [14] Duchênes, V., Vukicevic, I., and Weinstein, M., Scattering and localization properties of highly oscillatory potentials. Commun. Pure Appl. Math. 67(2014), no. 1, 83–128. http://dx.doi.Org/10.1OO2/cpa.21459 Google Scholar

[15] [15] Duchênes, V., Vukicevic, I., and Weinstein, M., Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes. SIAM J. Math. Anal. 47(2015), no. 5, 3832–3883. http://dx.doi.Org/10.1137/140980302 Google Scholar

[16] [16] Duchênes, V., Vukicevic, I., and Weinstein, M., Homogenized description of defect modes in periodic structures with localized defects. Commun. Math. Sci. 13(2015), no. 3, 777–823. Google Scholar | DOI

[17] [17] Duchênes, V. and Weinstein, M. Scattering, Homogenization, and Interface Effects for Oscillatory Potentials with Strong Singularities, Multiscale Mode Simul l. 9(2011), no. 3,1017-1063. http://dx.doi.Org/10.1137/100811 672 Google Scholar

[18] [18] Gerard, C., Martinez, A., and Sjostrand, J., A mathematical approach to the effective hamiltonian in perturbed periodic problems. Comm. Math. Phys. 142(1991), no. 2, 217–244. http://dx.doi.Org/10.1007/BF02102061 Google Scholar

[19] [19] Gerard, C. and Nier, F., Scattering theory for the perturbations of periodic Schrodinger operators. J. Math. Kyoto Univ. 38(1998), 595–634. Google Scholar

[20] [20] Guillot, J-C., Ralston, J., and Trubowitz, E., Semiclassical methods in solid state physics. Comm. Math. Phys. 116(1988), no. 3, 401–415. Google Scholar | DOI

[21] [21] Kato, T., Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, New York, 1966. Google Scholar

[22] [22] Laptev, A., Spectral inequalities for Partial Differential Equations and their applications. Proceedings of ICCM2010 in Beijing, AMS/IP Studies in Advanced Mathematics, 51, pt.2, American Mathematical Society, Providence, RI, 2012, pp. 629–643. Google Scholar

[23] [23] Laptev, A., Naboko, S., and Safronov, O., Absolutely continuous spectrum of Schrodinger operators with slowly decaying and oscillating potentials. Comm. Math. Phys. 253(2005), no. 3, 611–631. Google Scholar | DOI

[24] [24] Parnovski, L. and Shterenberg, R., Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrodinger operators. Duke Math. J. Volume 165(2016), no. 3, 509–561. http://dx.doi.Org/10.125/00127094-3166415 Google Scholar

[25] [25] Raikov, G., Discrete spectrum for Schrodinger operators with oscillating decaying potentials. J. Math. Anal. Appl. 438(2016), no. 2, 551–564. http://dx.doi.Org/10.1016/j.jmaa.201 6.02.005 Google Scholar

[26] [26] Reed, M. and Simon, B., Methods of modern mathematical physics. IV. Academic Press, New York, 1978. Google Scholar

[27] [27] Robert, D., Autour de I'approximation semi-classique. Progress in Mathematics, 68, Birkhauser Boston, Inc., Boston, MA., 1987. Google Scholar

[28] [28] Rozenblum, G. V., The distribution of the discrete spectrum of singular differential operators. English transl.: Sov. Math. Izv. VUZ 20(1976), 63–71. Google Scholar

[29] [29] Skriganov, M. M., The spectrum band structure of the three-dimensional Schrodinger operator with periodic potential. Invent. Math. 80(1985), no. 1,107-121. http://dx.doi.Org/10.1007/BF01388550 Google Scholar

[30] [30] Skriganov, M. M., Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. (Russian) Trudy Mat. Inst. Steklov. 171(1985). Google Scholar

[31] [31] Sjostrand, J., Microlocal analysis for the periodic magnetic Schrodinger equation and related questions. In: Microlocal analysis and applications (Montecatini Terme, 1989), Lecture Notes in Math., 1495, Springer, Berlin, 1991, pp. 237–332. Google Scholar | DOI

[32] [32] Shubin, M. A., The spectral theory and the index of elliptic operators with almost periodic coefficients. Russian Math. Surveys 34(1979), no. 2,109-158. Google Scholar

[33] [33] Suslina, T. A., Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. II. Internal gaps. Algebra i Analiz 15(2003), no. 2,128-289; St. Petersburg Math. J. 15(2004), 249–287. Google Scholar | DOI