• 论文 •

### 无序体系薛定谔算子格林函数及谱分布

1. 北京师范大学数学科学学院, 北京 100875
• 收稿日期:2020-11-12 发布日期:2021-03-18
• 通讯作者: 阙嘉豪,Email:quejiahao@live.com.
• 基金资助:
国家级大学生创新创业训练计划.

Chen Xuanwei, Que Jiahao, Wang Xinyue. GREEN'S FUNCTION AND SPECTRAL DISTRIBUTION OF SCHRODINGER OPERATOR FOR DISORDERED SYS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(1): 56-70.

### GREEN'S FUNCTION AND SPECTRAL DISTRIBUTION OF SCHRODINGER OPERATOR FOR DISORDERED SYS

Chen Xuanwei, Que Jiahao, Wang Xinyue

1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
• Received:2020-11-12 Published:2021-03-18

Disordered systems usually have unique optoelectronic properties. With the development of material application, disordered systems attracted more and more attention. The most important physical property of disordered systems is its electronic properties, which are related to the properties of the Schrödinger operator spectrum. The key to study the properties of Schrödinger operator spectrum is the attenuation of Green's function. It can describe the state of microscopic particles mathematically, and it is of great significance to explain and predict corresponding physical phenomena. According to the attenuation properties of Schrödinger operator Green's function of two typical disordered systems and related corollaries of spectral distribution, it can be seen that two operators are similar, but current research methods of these are different. In the future, we can seek more unified research methods for the localization of eigenfunctions of quasi-periodic systems, stochastic systems and other disordered systems, and extend them to more general disordered systems.

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 [1] Aizenman M and Molchanov S. Localization at large disorder and at extreme energies:an elementary derivation[J]. Comm. Math. Phys., 1993, 157:245-278.[2] Aizenman M, Schenker J H, Friedrich R M and Hundertmark D. Finite-volume fractional-moment criteria for Anderson localization[J]. Comm. Math. Phys., 2001, 224:219-253.[3] Aizenman M and Graf G M. Localization bounds for an electron gas[J]. J. Phys. A., 1998, 31:6783-6806.[4] Avila A and Jitomirskaya S. The ten Martini problem[J]. Annals of Mathematics, 2009, 170:303-342.[5] Bloch F.Über die quantenmechanik der elektronen in kristallgittern[J]. Z. Phys., 1928, 52:555.[6] Boosting superconductivity, Chemical & Engineering News, 2000, 78:38.[7] Bourgain J and Kenig C E. On localization in the continuous Anderson-Bernoulli model in higher dimension[J]. Invent. Math., 2005, 161:389-426.[8] Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C and Jarillo-Herrero P. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices[J]. Nature, 2018, 556:80-84.[9] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E and Jarillo-Herrero P. Unconventional superconductivity in magic-angle graphene superlattices[J]. Nature, 2018, 556:43-50.[10] Carmona R, Klein A and Martinelli F. Anderson localization for Bernoulli and others ingular potentials[J]. Comm. Math. Phys., 1987, 108:41-66.[11] Casdagli M. Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation[J]. Comm. Math. Phys., 1986, 107:295-318.[12] Choi M, Elliott G A and Yui N. Gauss polynomials and the rotation algebra, Invent Math., 1990, 99:225-246.[13] Damanik D, Embree M and Gorodetski A. Spectral properties of Schrödinger operators arising in the study of quasicrystals[J]. Mathematics of Aperiodic Order (J. Kellendonk, D. Lenz and J. Savinien, eds.), Progress in Mathematics, Birkhöuser, Basel, 2015, 309:307-370.[14] Damanik D, Gorodetski A and Yessen W. The Fibonacci Hamiltonian[J]. Inventiones Mathematicae, 2016, 206(3):629-692.[15] Damanik D. Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals[J]. American Mathematical Society, Providence, 2000, 277-305.[16] Damanik D. Schrödinger operators with dynamically dened potentials:A survey, in preparation.[17] Damanik D. Strictly ergodic subshifts and associated operators, in Spectral Theory and Mathematical Physics:a Festschrift in Honor of Barry Simon's 60th Birthday, 505-538, Proc. Sympos. Pure Math. 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.[18] delRio R, Jitomirskaya S, Last Y and Simon B. What is localization?[J]. Phys. Rev. Lett., 1995, 75:117-119.[19] delRio R, Jitomirskaya S, Last Y and Simon B. Operators with singular countinuous spectrum.IV.Hausdorff dimensions, rank one perturbations and localization[J]. J. Anal. Math., 1996, 69:153-200.[20] Denisov S and Kiselev A. Spectral properties of Schrödinger operators with decaying potentials. Spectral Theory and Mathematical Physics:a Festschrift in Honor of Barry Simon's 60th Birthday, 565-589, Proc. Sympos. Pure Math. 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.[21] Dinaburg E I and Sinai Ya G. The one-dimensional Schrödinger equation with a quasi-periodic potential[J]. Functional Anal. Appl., 1975, 9:279.[22] Dreifus H V and Klein A. A new proof of localization in the Anderson tight binding model[J]. Communications in Mathematical Physics, 1989, 124:285-299.[23] Elliott G A. Thoughts on the dry ten Martini problem. http://www.fields.utoronto.ca/programs/scientific/13-14/COSY2014, 2014, Lecture note.[24] Hundertmark D. A short introduction to Anderson localization. Analysis and Stochastics of Growth Processes and Interface Models. Oxford University Press. 2007, 194.[25] Jitomirskaya S Y. Anderson localization for the almost Mathieu equation:A nonperturbative proof[J]. Communications in Mathematical Physics, 1994, 165:49-57.[26] Jitomirskaya S. Ergodic Schrödinger operators (on one foot), in Spectral Theory and Mathematical Physics:a Festschrift in Honor of Barry Simon's 60th Birthday, 613-647, Proc. Sympos. Pure Math. 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.[27] Jitomirskaya S Y. Metal-insulator transition for the almost Mathieu operator[J]. Annals of Mathematics, 1999, 150:1159-1175.[28] Jitomirskaya S and Marx C. Dynamics and spectral theory of quasi-periodic Schrödinger type operators. in preparation.[29] Kohmoto M, Kadanoff L P and Tang C. Localization problem in one dimension:Mapping and escape[J]. Phys. Rev. Lett., 1983, 50:1870-1876.[30] Li X and Zhu H. Two-dimensional MoS2:Properties, preparation, and applications[J]. Journal of Materiomics, 2015, 1:33-44.[31] Liu W and Yuan X. Spectral gaps of almost Mathieu operator in exponential regime[J]. J. Fractal Geom., 2015, 2:1-51.[32] Ostlund S, Pandit R, Rand D, Schellnhuber H and Siggia E. One-dimensional Schrödinger equation with an almost periodic potential[J]. Phys. Rev. Lett., 1983, 50:1873-1877.[33] Peierls R. Zur theorie des diamagnetismus von leitungselektronen[J]. Z. Phys., 1933, 80:763-791.[34] Simon B and Wolff T. Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians[J]. Comm. PureAppl. Math., 1986, 39:75-90.[35] Simon B. Szegö's theorem and its descendants. spectral theory for L2 perturbations of orthogonal polynomials[M]. Princeton University Press, Princeton, 2011.[36] Spencer T. The Schrodinger equation with a random potential:A mathematical review[M]. Lecture Notes, Les Rouches Summer School, 1984.[37] Surace S. The Schrödinger equation with a quasi-periodic potential[J]. Transactions of the American Mathematical Society, 1990, 320:321-370.[38] Sütö A. The spectrum of a quasiperiodic Schrödinger operator[J]. Commun. Math. Phys., 1987, 111:409-415.[39] 阎守胜. 固体物理基础[M]. 北京:北京大学出版社, 2000, 126-129.
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