## Abstract

The survey reviews recent results on spectral analysis of differential and finite-difference operators with random spatially homogeneous coefficients. The corresponding problems that crystallized in the development of a number of areas in mathematics and related sciences are very rich and diverse. We discuss the traditional problems of spectral analysis, where the use of probabilistic ideas and methods now allows highly detailed spectral analysis to be performed for an essentially broader class of operators, as well as new problems and results obtained in the framework of this theory.

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## Literature cited

### Publications in Russian and Russian translations

- 1.
F. Atkinson, Discrete and Continuous Boundary Problems [Russian translation], Mir, Moscow (1968).

- 2.
E. D. Belokolos, “A quantum particle in a one-dimensional deformed lattice,” Teor. Mat. Fiz.,

__25__, 344–358 (1975). - 3.
M. M. Benderskii and L. A. Pastur, “Computing the mean number of states in one typical problem,” Zh. Éksp. Teor. Fiz.,

__57__, No. 7, 284–294 (1969). - 4.
M. M. Benderskii and L. A. Pastur, “On the spectrum of the one-dimensional Schrödinger equation with random potential,” Mat. Sb.,

__82__, 273–284 (1970). - 5.
M. M. Benderskii and L. A. Pastur, “On the asymptotic solutions of the equation of second order with random coefficients,” Teor. Funktsii, Funktsional'nyi Anal. Prilozhen., No. 22, 3–14 (1975).

- 6.
Yu. M. Berezanskii, Decomposition in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).

- 7.
T. E. Bogorodskaya, “Density of states of multidimensional integral operators,” Funkts. Anal. Prilozh.,

__13__, No. 2, 79–80 (1979). - 8.
T. E. Bogorodskaya and M. A. Shubin, “Variational principle and asymptotic behavior of the density of states for random pseudodifferential operators,” Tr. Seminara im. I. G. Petrovskogo,

__14__, 98–117 (1986). - 9.
V. L. Bonch-Bruevich, I. P. Zvyagin, R. Kuiper, A. G. Mironov, R. Énderline, and B. Ésser, Electronic Theory of Disordered Semiconductors [in Russian], Nauka, Moscow (1981).

- 10.
B. F. Bylov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytskii, The Theory of Lyapunov Exponents [in Russian], Nauka, Moscow (1966).

- 11.
V. K. Vardazaryan, “Spectral theory of a one-dimensional random Dirac system,” Dokl. AN ArmSSR,

__64__, No. 5, 264–270 (1977). - 12.
A. D. Ventsel' and M. I. Freidlin, Fluctuations in Dynamical Systems under the Action of Small Random Perturbations [in Russian], Nauka, Moscow (1979).

- 13.
V. S. Videnskii, “Entire transcendental N-functions and their application to the study of N-functions,” Mat. Sb.,

__62__, No. 2, 121–139 (1963). - 14.
A. D. Virtser, “On matrix and operator products,” Teor. Veroyatn. Primen.,

__24__, No. 2, 360–370 (1979). - 15.
V. L. Girko, Random Matrices [in Russian], Vishcha Shkola, Kiev (1975).

- 16.
I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes [in Russian], Vols. 1–3, Nauka, Moscow (1971–1975).

- 17.
I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], GIFML, Moscow (1963).

- 18.
I. Ya. Gol'dsheidt, “The law of large numbers in some functional spaces,” Usp. Mat. Nauk,

__31__, No. 2, 211–212 (1976). - 19.
I. Ya. Gol'dsheidt, “The structure of the spectrum of a random difference Schrödinger operator,” Dokl. AN SSSR,

__255__, No. 2, 273–277 (1980). - 20.
I. Ya. Gol'dsheidt, S. A. Molchanov, and L. A. Pastur, “A typical one-dimensional Schrödinger operator has a pure point spectrum,” Funkts. Anal. Prilozhen.,

__11__, No. 1, 1–10 (1977). - 21.
A. Ya. Gordon, “On the continuous spectrum of the one-dimensional Schrödinger operator,” Funkts. Anal. Prilozhen.,

__13__, No. 3, 77–78 (1979). - 22.
L. D. Gor'kov, O. N. Dorokhov, and F. V. Prigara, “The structure of wave functions and ac conductivity in one-dimensional disordered semiconductors,” Zh. Éksp. Teor. Fiz.,

__85__, No. 4, 1470–1488 (1983). - 23.
L. D. Gor'kov and G. M. Éliashberg, “Small metallic particles in an electromagnetic field,” Zh. Éksp. Teor. Fiz.,

__48__, No. 4, 1407–1418 (1965). - 24.
S. A. Gredeskul and L. A. Pastur, “On the behavior of density of states in one-dimensional disordered systems near the spectrum boundaries,” Teor. Mat. Fiz.,

__23__, No. 1, 132–139 (1975). - 25.
S. A. Gredeskul and L. A. Pastur, “Density of states in a one-dimensional disordered system in a two-band approximation,” Zh. Éksp. Teor. Fiz.,

__75__, 1444–1459 (1978). - 26.
S. A. Gredeskul and L. A. Pastur, “Density of states near the fluctuation boundary of the spectrum of one-dimensional incommensurate structures,” Teor. Mat. Fiz.,

__62__, No. 2, 316–319 (1985). - 27.
L. N. Grenkova, “On essential self-adjointness of the Schrödinger operator with random potential,” Usp. Mat. Nauk,

__36__, No. 6, 211–212 (1981). - 28.
L. N. Grenkova, S. A. Molchanov, and Yu. I. Sudarev, “The structure of energy levels and wave functions near the edge of the spectrum of one-dimensional disordered structures,” Dokl. AN SSSR,

__263__, No. 3, 576–580 (1982). - 29.
A. I. Gusev, “State density and other spectral invariants of self-adjoint elliptical operators with random coefficients,” Mat. Sb.,

__104__, No. 2, 207–226 (1977). - 30.
F. Dyson, Level Statistics of Complex Nuclei [Russian translation], Mir, Moscow (1960).

- 31.
E. I. Dinaburg and Ya. G. Sinai, “On one-dimensional Schrödinger equation with quasiperiodic potential,” Funkts. Anal. Prilozh.,

__9__, No. 4, 8–21 (1975). - 32.
A. M. Dykhne, “Conductivity in a two-dimensional two-phase system,” Zh. Éksp. Teor. Fiz.,

__59__, No. 7, 110–115 (1970). - 33.
V. V. Dyakin and S. I. Petrukhnovskii, “Some geometrical properties of Fermi surfaces,” Dokl. AN SSSR,

__264__, No. 5, 1117–1119 (1982). - 34.
G. M. Zaslavskii, Stochastic Dynamical Systems [in Russian], Nauka, Moscow (1984).

- 35.
I. S. Kats, “Multiplicity of the spectrum of a differential operator of second order and expansion in eigenfunctions,” Izv. AN SSSR, Ser. Mat.,

__27__, 1081–1112 (1963). - 36.
S. M. Kozlov, “Conductivity of two-dimensional random media,” Usp. Mat. Nauk,

__34__, No. 4, 193–194 (1979). - 37.
S. M. Kozlov, “Averaging of random operators,” Mat. Sb.,

__109__, No. 2, 188–202 (1979). - 38.
S. M. Kozlov, “The method of averaging and random walk in inhomogeneous media,” Usp. Mat. Nauk,

__40__, No. 2, 61–120 (1985). - 39.
I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).

- 40.
L. D. Landau and E. A. Smorodinskii, Lectures in the Theory of Atomic Nucleus [in Russian], GTTI, Moscow (1957).

- 41.
B. M. Levitan, Almost-Periodic Functions [in Russian], GITTL, Moscow (1953).

- 42.
B. M. Levitan and A. V. Savin, “Examples of Schrödinger operators with almost-periodic potentials and nowhere dense absolutely continuous spectrum,” Dokl. AN SSSR,

__276__, No. 3, 539–542 (1984). - 43.
B. M. Levitan and I. S. Sargsyan, An Introduction to Spectral Theory [in Russian], Nauka, Moscow (1970).

- 44.
I. M. Lifshits, “On the structure of the energy spectrum and quantum state of disordered condensed systems,” Usp. Fiz. Nauk,

__83__, 617–663 (1964). - 45.
I. M. Lifshits, “The theory of fluctuation levels in disordered systems,” Zh. Éksp. Teor. Fiz.,

__53__, 743–758 (1967). - 46.
I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, “To the theory of propagation of particles and waves through randomly inhomogeneous media,” Zh. Éksp. Teor. Fiz.,

__83__, No. 6, 2362–2367 (1982). - 47.
I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, An Introduction to the Theory of Disordered Systems [in Russian], Nauka, Moscow (1982).

- 48.
V. A. Marchenko, Spectral Theory of Sturm-Liouville Operators [in Russian], Naukova Dumka, Kiev (1977).

- 49.
V. A. Marchenko and L. A. Pastur, “Distribution of eigenvalues in some ensembles of random matrices,” Mat. Sb.,

__72__, No. 4, 507–536 (1967). - 50.
V. M. Millionshchikov, “Statistically regular systems,” Mat. Sb.,

__75__, No. 1, 140–151 (1968). - 51.
R. A. Minlos and A. Ya. Povzner, “On the thermodynamic limit for entropy,” Trudy Moskovsk. Mat. Obshch.,

__17__, 243–272 (1967). - 52.
S. A. Molchanov, “The structure of eigenfunctions of one-dimensional disordered structures,” Izv. AN SSSR, Ser. Math.,

__42__, No. 11, 70–103 (1978). - 53.
S. A. Molchanov, A. A. Ruzmaikin, and D. D. Sokolov, “The kinematic dynamo in a random flow,” Usp. Fiz. Nauk,

__145__, No. 4, 593–628 (1985). - 54.
S. A. Molchanov and A. K. Stepanov, “Bursts of a Gaussian field above a high level,” Dokl. AN SSSR,

__249__, No. 2, 294–297 (1979). - 55.
S. A. Molchanov and V. N. Tutubalin, “A linear model of the hydrodynamic dynamo and products of random matrices,” Teor. Veroyatn. Primen.,

__29__, No. 2, 234–242 (1984). - 56.
V. I. Oseledets, “Multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems,” Trudy Mosk. Mat. Obshch.,

__17__, 179–210 (1968). - 57.
L. A. Pastur, “On Schrödinger equation with random potential,” Teor. Mat. Fiz.,

__6__, No. 3, 415–424 (1971). - 58.
L. A. Pastur, “On the spectrum of random matrices,” Teor. Mat. Fiz.,

__10__, No. 1, 102–112 (1972). - 59.
L. A. Pastur, “On the distribution of eigenvalues of the Schrödinger equation with random potential,” Funkts. Anal. Prilozh.,

__6__, No. 1, 93–94 (1972). - 60.
L. A. Pastur, “Spectra of random self-adjoint operators,” Usp. Mat. Nauk,

__28__, No. 1, 3–64 (1973). - 61.
L. A. Pastur, “On the spectrum of random Jacobi matrices and the Schrödinger operator on the entire axis with random potential,” Preprint, Fiz.-Tekh. Inst. Nizkikh Temp. AN UkrSSR, Kharkov (1974).

- 62.
L. A. Pastur, “On the distribution of eigenvalues of the Schrödinger equation with random potential,” in: Mathematical Physics, Functional Analysis [in Russian], Vol. 5, FTINT AN UkrSSR, Kharkov (1974), pp. 141–143.

- 63.
L. A. Pastur, “The behavior of some Wiener integrals for t→∞ and the density of states of the Schrödinger equation with random potential,” Teor. Mat. Fiz.,

__32__, No. 1, 88–95 (1977). - 64.
L. A. Pastur and V. A. Tkachenko, “To the spectral theory of one-dimensional Schrödinger operator with limit-periodic potential,” Dokl. AN SSSR,

__274__, No. 5, 1150–1153 (1984). - 65.
L. A. Pastur and É. P. Fel'dman, “On the transmission coefficient of a wave through a thick layer of a randomly inhomogeneous medium,” Zh. Éksp. Teor. Fiz.,

__67__, No. 2, 487–493 (1974). - 66.
L. A. Pastur and A. L. Figotin, “Ergodic properties of the distribution of eigenvalues of some classes of random self-adjoint operators,” in: Differential Equations and Some Methods of Functional Analysis [in Russian], Naukova Dumka, Kiev (1978), pp. 117–133.

- 67.
A. Ya. Reznikova, “The central limit theorem for spectra of random Jacobi matrices,” Teor. Veroyatn. Primen.,

__25__, No. 3, 513–522 (1980). - 68.
M. Reed and B. Simon, Methods of Modern Mathematical Physics [Russian translation], Vols. 1–4, Mir, Moscow (1977).

- 69.
D. Ruelle, Statistical Mechanics [Russian translation], Mir, Moscow (1971).

- 70.
V. N. Tutubalin, “On limit theorems for products of random matrices,” Teor. Veroyatn. Primen.,

__10__, No. 1, 19–32 (1965). - 71.
B. V. Fedosov and M. A. Shubin, “The index of random elliptical operators,” Mat. Sb.,

__106__, No. 1, 108–140; No. 3, 455–483 (1978). - 72.
W. Feller, An Introduction to Probability Theory and Its Applications [Russian translation], Mir, Moscow (1964).

- 73.
A. L. Figotin, “Large-time asymptotic behavior of some Wiener integrals,” Teor. Funktsii, Punktsional'nyi Analiz Prilozhen.,

__32__, 88–91 (1979). - 74.
A. L. Figotin, “On exponential growth of the solutions of the difference equation of second order with random coefficients,” Dokl. AN UzSSR, No. 2, 9–10 (1980).

- 75.
A. L. Figotin, “Distribution of the eigenvalues of the Schrödinger equation with random potential and large-time asymptotic behavior of one Wiener process,” Dokl. AN UkrSSR, Ser. A, No. 6, 27–29 (1981).

- 76.
A. L. Figotin, “Essential self-adjointness and ergodic properties of the Schrödinger operator with random potential,” Dokl. AN UkrSSR, Ser. A, No. 8, 18–20 (1983).

- 77.
P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).

- 78.
R. Z. Khas'minskii, Stability of Systems of Differential Equations with Randomly Perturbed Parameters [in Russian], Nauka, Moscow (1969).

- 79.
V. A. Chulaevskii, “The inverse spectral problem for limit-periodic Schrödinger operators,” Funkts. Anal. Prilozhen.,

__18__, No. 3, 63–66 (1984). - 80.
B. I. Shklovskii and A. L. Éfros, Electronic Properties of Alloyed Semiconductors [in Russian], Nauka, Moscow (1979).

- 81.
I. É. Shnol', “The behavior of eigenfunctions and the spectrum of Sturm-Liouville operators,” Usp. Mat. Nauk,

__9__, No. 4, 389–392 (1954). - 82.
M. A. Shubin, “Elliptical almost-periodic operators and von Neumann algebras,” Funkts. Anal. Prilozhen.,

__9__, No. 1, 89–90 (1975). - 83.
M. A. Shubin, “Spectral theory and the index of elliptical operators with almost periodic coefficients,” Usp. Mat. Nauk,

__34__, No. 2, 95–136 (1979).

### Publications in other languages

- 84.
V. V. Anshelevich, K. M. Khanin, and Ya. G. Sinai, “Symmetric random walks in random environments,” Commun. Math. Phys.,

__85__, 449 (1982). - 85.
L. Arnold and V. Wihstutz (eds.), Lyapunov Exponents, Lecture Notes Math., Vol. 1186, Springer (1986).

- 86.
J. Avron and B. Simon, “Singular continuous spectrum for a class of almost periodic Jacobi matrices,” Bull. AMS,

__6__, No. 1, 81–87 (1982). - 87.
J. Belissard, D. Bessis, and P. Moussa, “Chaotic states of almost periodic Schrödinger operators,” Phys. Rev. Lett.,

__49__, No. 10, 701–704 (1982). - 88.
J. Belissard, R. Lima, and E. Scoppola, “Localization in υ-dimensional incommensurate structures,” Commun. Math. Phys.,

__88__, 465–477 (1983). - 89.
F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, and R. Weder, “Schrödinger operators with electric field and random or deterministic potentials,” Commun. Math. Phys.,

__88__, No. 4, 387–397 (1983). - 90.
R. Borland, “The nature of the electronic states in disordered one-dimensional systems,” Proc. R. Soc. London,

__A274__, No. 1359, 529–540 (1963). - 91.
P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhauser, Boston (1985).

- 92.
M. Campanino and A. Klein, “A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson mode,” Commun. Math. Phys.,

__104__, No. 2, 227–241 (1986). - 93.
R. Carmona, “Exponential localization in one-dimensional disordered systems,” Duke Math. J.,

__49__, No. 1, 191–213 (1982). - 94.
R. Carmona, “One-dimensional Schrödinger operators with random potentials,” Physica,

__124A__, 181–188 (1984). - 95.
R. Carmona, “Random Schrödinger operators,” Lecture Notes Math., Vol. 1180, 2–124 (1980).

- 96.
R. Carmona, A. Klein, and F. Martinelli, “Anderson localization for Bernoulli and other singular potentials,” Preprint, Univ. Calif., Irvine (1986).

- 97.
W. Craig and B. Simon, “Log Hoelder continuity of the integrated density of states for stochastic Jacobi matrices,” Commun. Math. Phys.,

__90__, No. 2, 207–218 (1983). - 98.
W. Craig and B. Simon, “Subharmonicity of the Lyapunov index,” Duke Math J.,

__50__, No. 4, 551–560 (1983). - 99.
P. Deift and B. Simon, “Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension,” Commun. Math. Phys.,

__90__, 389–411 (1983). - 100.
P. Delyon, H. Kunz, and B. Souillard, “One-dimensional wave equation in disordered media,” J. Phys.,

__A16__, No. 1, 25–40 (1983). - 101.
F. Delyon, B. Simon, and B. Souillard, “From power pure point to continuous spectrum in disordered systems,” Ann. Inst. H. Poincare,

__42__, No. 3, 283–309 (1985). - 102.
F. Delyon and B. Souillard, “Remark on the continuity of the density of states of ergodic finite-difference operators,” Commun. Math. Phys.,

__94__, No. 2, 289–291 (1984). - 103.
F. Delyon, Y. Levy, and B. Souillard, “Anderson localization for quasi one-dimensional systems,” J. Stat. Phys.,

__41__, No. 3/4, 375–388 (1985). - 104.
F. Delyon, Y. Levy, and B. Souillard, “Anderson localization for multidimensional systems at large disorder or low energy,” Commun. Math. Phys.,

__100__, No. 4, 463–470 (1985). - 105.
F. J. Dyson, “The dynamics of a disordered linear chain,” Phys. Rev.,

__92__, No. 3, 1331–1338 (1953). - 106.
M. S. Eastham, W. D. Evans, and J. S. Mclead, “The essential self-adjointness of Schrödinger-type operators,” Arch. Rat. Mech. Anal.,

__60__, No. 1, 185–204 (1976). - 107.
K. V. Efetov, “Supersymmetry and theory of disordered systems,” Adv. Phys.,

__32__, No. 1, 53–127 (1983). - 108.
H. Ehglisch and K.-D. Kursten, “Infinite representability of Schrödinger operators with ergodic potential,” Z. Anal. Anwend.,

__2__, No. 3, 411–420 (1983). - 109.
A. L. Figotin and L. A. Pastur, “The positivity of the Lyapunov exponent and the absence of the absolutely continuous spectrum for the almost-Mathieu equation,” J. Math. Phys.,

__25__, No. 4, 774–777 (1984). - 110.
A. L. Figotin and L. A. Pastur, “An exactly solvable model of a multidimensional incommensurable structure,” Commun. Math. Phys.,

__95__, No. 4, 401–425 (1984). - 111.
J. Frohlich and T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys.,

__88__, 151–189 (1983). - 112.
J. Frohlich, F. Martinelli, E. Scoppola, and T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys.,

__101__, No. 1, 21–46 (1985). - 113.
F. Fukushima, “On the spectral distribution of a disordered system and the range of a random walk,” Osaka J. Math.,

__11__, No. 1, 73–85 (1974). - 114.
H. Furstenberg, “Noncommuting random products,” Trans. AMS,

__108__, No. 2, 377–428 (1963). - 115.
D. Herbert and R. Jones, “Localized states in disordered systems,” J. Phys.,

__C4__, No. 10, 1145–1150 (1971). - 116.
M. Herman, “Une methode pour minorer les exposants de Lyapunov,” Comment. Math. Helv.,

__58__, 453–502 (1983). - 117.
H. Holden and F. Martinelli, “Absence of diffusion near the bottom of the spectrum for a Schrödinger operator on L2(R)

^{+},” Commun. Math. Phys.,__93__, 197–217 (1984). - 118.
L. Hormander, “Hypoelliptic differential equations of second order,” Acta Math.,

__119__, No. 1, 147–171 (1967). - 119.
K. Ichihara and H. Kunita, Z. Wahrschein, Verw. Gebiete,

__30__, No. 2, 235–254 (1974). - 120.
K. Ishii, “Localization of eigenstates and transport phenomena in one-dimensional disordered systems,” Prog. Theor. Phys. Supp.,

__53__, 77 (1973). - 121.
G. Jona-Lasinio, F. Martinelli, and E. Scoppola, “A quantum particle in a hierarchical potential with tunneling over arbitrarily large scales,” Ann. Inst. H. Poincaré,

__42__, No. 1, 73–108 (1985). - 122.
W. Kirsch, S. Kotani, and B. Simon, “Absence of absolutely continuous spectrum for some one-dimensional random but deterministic Schrödinger operators,” Ann. Inst. H. Poincaré,

__42__, No. 4, 383–406 (1985). - 123.
W. Kirsch, “Random Schrödinger operators and the density of states,” Lecture Notes Math.,

__1190__, 69–102 (1985). - 124.
W. Kirsch and F. Martinelli, “Large deviations and Lifshitz-singularity of the integrated density of states of random Hamiltonians,” Commun. Math. Phys.,

__89__, 27–40 (1983). - 125.
A. Klein, F. Martinelli, and J. F. Perez, “A rigorous replica trick approach to Anderson localization in one dimension,” Preprint, Univ. Calif., Irvine (1986).

- 126.
S. Kotani, “Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators,” in: Proc. Taniguchi Symp. Stochastic Analysis, North-Holland, Amsterdam (1982), pp. 225–247.

- 127.
S. Kotani, “Support theorems for random Schrödinger operators,” Commun. Math. Phys.,

__97__, No. 3, 443–452 (1984). - 128.
S. Kotani, “On an inverse problem for random Schrödinger operators,” AMS Series of Contemp. Math., Providence, RI,

__41__, 267–280 (1985). - 129.
S. Kotani, “Lyapunov exponents and spectra for one-dimensional random Schrödinger operators,” AMS Series of Contemp. Math., Providence, RI,

__50__, 277–286 (1986). - 130.
H. Kunz and B. Souillard, “Sur le spectre des operateurs aux differences finies aleatoires,” Commun. Math. Phys.,

__79__, No. 2, 201–246 (1980). - 131.
P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys.,

__52__, No. 2, 287–337 (1985). - 132.
E. Le Page, “Repartition d'etats pour les matrices de Jacobi à coefficients aleatoires,” Lecture Notes Math.,

__1064__, 209–267 (1983). - 133.
F. Martinelli and E. Scoppola, “Remark on the absence of absolutely continuous spectrum in the Anderson tight binding model,” Commun. Math. Phys.,

__97__, 465–471 (1985). - 134.
F. Martinelli and E. Scoppola, Introduction to the Mathematical Theory of Anderson Localization, Preprint, Univ. La Sapienza, Roma (1986).

- 135.
H. Matsuda and K. Ishii, “Localization of normal modes and energy transport in the disordered harmonic chain,” Progr. Theor. Phys. Suppl.,

__45__, 56 (1970). - 136.
M. L. Mehtà, Random Matrices and the Statistical Theory of Energy Levels, Academic Press, New York (1967).

- 137.
G. A. Mezincescu, “Integral Lifshitz singularities of disordered finite-difference Schrödinger operators,” Commun. Math. Phys.,

__103__, No. 1, 167–176 (1986). - 138.
S. A. Molcanov [Molchanov], “The local structure of the spectrum of the one-dimensional Schrödinger operator,” Commun. Math. Phys.,

__78__, 429–446 (1981). - 139.
S. A. Molcanov [Molchanov] and B. Seidel, “Spectral properties of the general Sturm-Liouville operator with random coefficients, I,” Math. Nachr.,

__109__, 57–78 (1982). - 140.
J. Morrison, “On the number of electronic levels in a one-dimensional random lattice,” J. Math. Phys.,

__3__, No. 5, 1023–1028 (1962). - 141.
N. F. Mott and Q. D. Twose, “The theory of impurity conduction,” Adv. Phys.,

__10__, No. 1, 107–163 (1961). - 142.
T. M. Nieuwenhuizen, “Exact electronic spectra and inverse localization length in one-dimensional random systems,” Physica,

__120A__, No. 3, 468–514 (1983). - 143.
L. A. Pastur, “Spectral properties of disordered systems in the one-body approximation,” Commun. Math. Phys.,

__75__, 179–196 (1980). - 144.
L. A. Pastur, “Disordered spherical model,” J. Stat. Phys.,

__27__, No. 1, 119–151 (1982). - 145.
L. A. Pastur, “Spectral properties of random and almost periodic differential and finite-difference operators,” in: Statistical Physics and Dynamical Systems, Birkhauser, Boston (1985), pp. 49–67.

- 146.
L. A. Pastur, “On the pure point spectrum of the one-dimensional Anderson model with the Gaussian potential,” Preprint, Karl-Marx-Univ., Leipzig (1985).

- 147.
J. Piepenbrink and P. Rejto, “Some singular Schrödinger operators with deficiency indices (n

^{2}, n^{2}),” Duke Math. J.,__41__, No. 3, 593–605 (1974). - 148.
M. Romerio and W. Wreszinski, “On the Lifshitz singularity and the tailing in the density of states for random lattice systems,” J. Stat. Phys.,

__21__, No. 2, 169–179 (1979). - 149.
G. Royer, “Croissance exponentielle de produits markoviens de matrices aleatoires,” Ann. Inst. H. Poincaré,

__B16__, 49–62 (1980). - 150.
D. Ruelle, “A remark on bound states in potential scattering,” Nuovo Cimento,

__A61__, 655–661 (1969). - 151.
B. Simon, “Kotani theory for one-dimensional stochastic Jacobi matrices,” Commun. Math. Phys.,

__89__, No. 2, 227–236 (1983). - 152.
B. Simon, “Lifshitz trails for the Anderson model,” J. Stat. Phys.,

__38__, No. 1/2, 65–76 (1985). - 153.
B. Simon, “Almost periodic Schrödinger operators, IV, The Maryland model,” Ann. Phys. (USA),

__159__, No. 1, 157–183 (1985). - 154.
B. Simon, “Localization in general one-dimensional random systems, I, Jacobi matrices,” Commun. Math. Phys.,

__102__, No. 3, 327–336 (1985). - 155.
B. Simon, M. Taylor, and T. Wolff, “Some rigorous results for the Anderson model,” Phys. Rev. Lett.,

__54__, No. 14, 1589–1592 (1985). - 156.
L. Thomas, “Time dependent approach to scattering from impurities in crystals,” Commun. Math. Phys.,

__33__, No. 3, 335–343 (1973). - 157.
D. Thouless, “A relation between the density of states and range of localization for one-dimensional random system,” J. Phys.,

__C5__, No. 1, 77–81 (1972). - 158.
F. Wegner, “Bounds on the density of states in disordered systems,” Z. Phys.,

__B44__, 9–15 (1981). - 159.
E. Wigner, “Random matrices in physics,” SIAM Rev. J.,

__9__, No. 1, 1–23 (1967).

## Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 25, pp. 3–67, 1987.

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### Cite this article

Pastur, L.A. Spectral theory of random self-adjoint operators.
*J Math Sci* **46, **1979–2021 (1989). https://doi.org/10.1007/BF01096021

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### Keywords

- Spectral Analysis
- Recent Result
- Spectral Theory
- Probabilistic Idea
- Broad Class