Skip to main content

Products of Random Matrices and Generalised Quantum Point Scatterers

Abstract

To every product of 2×2 matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of random matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in SL(2,ℝ). We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  2. 2.

    Albeverio, S., Dabrowski, L., Kurasov, P.: Symmetries of the Schrödinger operators with point interactions. Lett. Math. Phys. 45, 33–47 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Springer-Verlag, Berlin (1988)

    MATH  Google Scholar 

  4. 4.

    Antsygina, T.N., Pastur, L.A., Slyusarev, V.A.: Localization of states and kinetic properties of one-dimensional disordered systems. Sov. J. Low Temp. Phys. 7, 1–21 (1981)

    Google Scholar 

  5. 5.

    Barnes, C., Luck, J.-M.: The distribution of the reflection phase of disordered conductors. J. Phys. A: Math. Gen. 23, 1717 (1990)

    Article  ADS  Google Scholar 

  6. 6.

    Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Birkhaüser, Basel (1985)

    MATH  Google Scholar 

  7. 7.

    Bouchaud, J.-P., Comtet, A., Georges, A., Le Doussal, P.: Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. (N.Y.) 201, 285–341 (1990)

    Article  ADS  Google Scholar 

  8. 8.

    Carmona, P., Petit, F., Yor, M.: On the distribution and asymptotic results for exponential functionals of Lévy processes. In: Yor, M. (ed.) Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, pp. 173–221. Madrid (1997)

  9. 9.

    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhaüser, Boston (1990)

    MATH  Google Scholar 

  10. 10.

    Chamayou, J.-F., Letac, G.: Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theor. Probab. 4, 3–36 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Cheon, T., Shigehara, T.: Continuous spectra of generalized Kronig-Penney model. J. Phys. Soc. Jpn. 73, 2986–2990 (2004)

    MATH  Article  ADS  Google Scholar 

  12. 12.

    Cheon, T., Shigehara, T.: Realizing discontinuous wave functions with renormalized short-range potentials. Phys. Lett. A 243, 111–116 (1998)

    Article  ADS  Google Scholar 

  13. 13.

    Chernoff, P.R., Hughes, R.J.: A new class of point interactions in one dimension. J. Funct. Anal. 111, 97–117 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    Comtet, A., Texier, C.: One-dimensional disordered supersymmetric quantum mechanics: a brief survey. In: Aratyn, H., et al. (eds.) Supersymmetry and Integrable Models. Lecture Notes in Physics, vol. 502, pp. 313–328. Springer, Berlin (1998). Also available as cond-mat/9707313

    Chapter  Google Scholar 

  15. 15.

    Cooper, F., Khare, A., Sukhatme, U.P.: Supersymmetry in Quantum Mechanics. World Scientific, Singapore (2001)

    MATH  Book  Google Scholar 

  16. 16.

    Deych, L.I., Lisyansky, A.A., Altshuler, B.L.: Single parameter scaling in 1-D Anderson localization. Exact analytical solution. Phys. Rev. B 64, 224202 (2001)

    Article  ADS  Google Scholar 

  17. 17.

    Exner, P.: Lattice Kronig-Penney models. Phys. Rev. Lett. 74, 3503 (1995)

    Article  ADS  Google Scholar 

  18. 18.

    Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York (1971)

    MATH  Google Scholar 

  19. 19.

    Frisch, H.L., Lloyd, S.P.: Electron levels in a one-dimensional lattice. Phys. Rev. 120, 1175–1189 (1960)

    MATH  Article  ADS  Google Scholar 

  20. 20.

    Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)

    MATH  MathSciNet  Google Scholar 

  21. 21.

    Gjessing, H.K., Paulsen, J.: Present value distributions with application to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  22. 22.

    Goldhirsch, I., Noskowicz, S.H., Schuss, Z.: Spectral degeneracy in the one-dimensional Anderson problem: a uniform expansion in the energy band. Phys. Rev. B 49, 14504–14522 (1994)

    Article  ADS  Google Scholar 

  23. 23.

    Gradstein, I.S., Ryzhik, I.M.: In: Jeffrey, A. (ed.) Tables of Integrals, Series and Products. Academic Press, New York (1964)

    Google Scholar 

  24. 24.

    Gurarie, V., Chalker, J.T.: Bosonic excitations in random media. Phys. Rev. B 68, 134207 (2003)

    Article  ADS  Google Scholar 

  25. 25.

    Halperin, B.I.: Green’s functions for a particle in a one-dimensional random potential. Phys. Rev. 139, A104–A117 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  26. 26.

    Herbert, D.C., Jones, R.: Localized states in disordered systems. J. Phys. C: Solid State Phys. 4, 1145 (1971)

    Article  ADS  Google Scholar 

  27. 27.

    Ishii, K.: Localization of eigenstates and transport phenomena in the one dimensional disordered system. Prog. Theor. Phys. (Suppl.) 53, 77–138 (1973)

    Article  ADS  Google Scholar 

  28. 28.

    Jayannavar, A.M., Vijayagovindan, G.V., Kumar, N.: Energy dispersive backscattering of electrons from surface resonances of a disordered medium and 1/f noise. Z. Phys., B Condens. Matter 75, 77 (1989)

    Article  ADS  Google Scholar 

  29. 29.

    Kotani, S.: On asymptotic behaviour of the spectra of a one-dimensional Hamiltonian with a certain random coefficient. Publ. RIMS, Kyoto Univ. 12, 447–492 (1976)

    Article  MathSciNet  Google Scholar 

  30. 30.

    de L. Kronig, R., Penney, W.G.: Quantum mechanics of electrons in crystal lattices. Proc. R. Soc. Lond. A 130, 499–513 (1931)

    MATH  Article  ADS  Google Scholar 

  31. 31.

    Lifshits, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the Theory of Disordered Systems. Wiley, New York (1988)

    Google Scholar 

  32. 32.

    Luck, J.-M.: Systèmes désordonnés unidimensionnels. Alea, Saclay (1992)

    Google Scholar 

  33. 33.

    Marklof, J., Tourigny, Y., Wolowski, L.: Explicit invariant measures for products of random matrices. Trans. Am. Math. Soc. 360, 3391–3427 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  34. 34.

    Nieuwenhuizen, T.M.: A new approach to the problem of disordered harmonic chains. Physica A 113, 173–202 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  35. 35.

    Nieuwenhuizen, T.M.: Exact electronic spectra and inverse localization lengths in one-dimensional random systems: I. Random alloy, liquid metal and liquid alloy. Physica A 120, 468–514 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  36. 36.

    Pollicott, M.: Maximal Lyapunov exponents for random matrix products. Invent. Math. (2010). doi:10.1007/S00222-004-0357-4

    Google Scholar 

  37. 37.

    Schomerus, H., Titov, M.: Band-center anomaly of the conductance distribution in one-dimensional Anderson localization. Phys. Rev. B 67, 100201 (2003)

    Article  ADS  Google Scholar 

  38. 38.

    S̆eba, P.: The generalized point interaction in one dimension. Czech. J. Phys. 36, 667–673 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  39. 39.

    Steiner, M., Chen, Y., Fabrizio, M., Gogolin, A.O.: Statistical properties of localization-delocalization transition in one dimension. Phys. Rev. B 59, 14848 (1999)

    Article  ADS  Google Scholar 

  40. 40.

    Texier, C., Comtet, A.: Universality of the Wigner time delay distribution for one-dimensional random potentials. Phys. Rev. Lett. 82, 4220–4223 (1999)

    Article  ADS  Google Scholar 

  41. 41.

    Texier, C., Hagendorf, C.: One-dimensional classical diffusion in a random force field with weakly concentrated absorbers. Europhys. Lett. 86, 37011 (2009)

    Article  ADS  Google Scholar 

  42. 42.

    Thouless, D.J.: A relation between the density of states and range of localization for one-dimensional random systems. J. Phys. C: Solid State Phys. 5, 77 (1972)

    Article  ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Christophe Texier.

Additional information

We thank Jean-Marc Luck for drawing to our attention the work of T.M. Nieuwenhuizen, and Tom Bienaimé for participating in the study of the supersymmetric model.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Comtet, A., Texier, C. & Tourigny, Y. Products of Random Matrices and Generalised Quantum Point Scatterers. J Stat Phys 140, 427–466 (2010). https://doi.org/10.1007/s10955-010-0005-x

Download citation

Keywords

  • Random matrices
  • Disordered one-dimensional quantum mechanics
  • Anderson localisation
  • Lyapunov exponent
  • Generalised point scatterers
  • Supersymmetric quantum mechanics