Communications in Mathematical Physics

, Volume 332, Issue 2, pp 827–838 | Cite as

A Note on Reflectionless Jacobi Matrices

  • V. Jakšić
  • B. Landon
  • A. Panati


The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of Breuer et al. (Commun Math Phys 295:531–550, 2010). The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection.


Jacobi Matrix Jacobi Matrice Jacobi Operator Diagonal Matrix Element Commun Math Phys 
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  1. BRS.
    Breuer J., Ryckman E., Simon B.: Equality of the spectral and dynamical definitions of reflection. Commun. Math. Phys. 295, 531–550 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. CLP.
    Chu, S., Landon, B., Panangaden, J.: In preparationGoogle Scholar
  3. DaSi.
    Davies E.B., Simon B.: Scattering theory for systems with different spatial asymptotics on the left and right. Commun. Math. Phys. 63, 277–301 (1978)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. DeSi.
    Deift P., Simon B.: Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys. 90, 398–411 (1983)ADSCrossRefMathSciNetGoogle Scholar
  5. GKT.
    Gesztesy F., Krishna M., Teschl G.: On isospectral sets of Jacobi operators. Commun. Math. Phys. 181, 631–645 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. GNP.
    Gesztesy F., Nowell R., Potz W.: One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics. Diff. Integral Eqs. 10, 521–546 (1997)zbMATHMathSciNetGoogle Scholar
  7. GS.
    Gesztesy F., Simon B.: Inverse spectral analysis with partial information on the potential, I. The case of an a.c. component in the spectrum. Helv. Phys. Acta 70, 66–71 (1997)zbMATHMathSciNetGoogle Scholar
  8. J.
    Jakšić, V.: Topics in spectral theory. In: Open quantum systems I. the hamiltonian approach. Attal, S., Joye, A., Pillet, C.-A. (eds). Lecture Notes in Mathematics 1880. Berlin: Springer, 2006Google Scholar
  9. JKP.
    Jakšić, V., Kritchevski, E., Pillet, C.-A.: Mathematical theory of the Wigner-Weisskopf atom. In: Large coulomb systems. Dereziński, J., Siedentop, H. (eds). Lecture Notes in Physics 695. Berlin: Springer, 2006Google Scholar
  10. JLP.
    Jakšić V., Landon B., Pillet C.-A.: Entropic fluctuations of XY quantum spin chains and reflectionless Jacobi matrices. Ann. Henri Poincaré 14, 1775–1800 (2013)ADSCrossRefzbMATHGoogle Scholar
  11. JOPP.
    Jakšić, V., Ogata, Y., Pautrat, Y., Pillet, C.-A.: Entropic fluctuations in quantum statistical mechanics—an introduction. In: Quantum theory from small to large scales. Fröhlich, J. Salmhofer, M. Mastropietro, V. De Roeck, W., Cugliandolo, L.F. (eds). Oxford: Oxford University Press, 2012Google Scholar
  12. La.
    Landon, B.: Master’s thesis. McGill University, 2013Google Scholar
  13. Si.
    Simon B.: Szegö’s Theorem and its Descendants. Spectral Theory for L 2 Perturbations of Orthogonal Polynomials. M. B. Porter Lectures. Princeton: Princeton University Press, 2011Google Scholar
  14. SY.
    Sodin M., Yuditskii P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. Te.
    Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs 72. Providence: AMS, 2000Google Scholar
  16. Y.
    Yafaev, D.R.: Mathematical Scattering Theory. General Theory. Translated from Russian by J. R. Schulenberger. Translations of Mathematical Monographs, 105. Providence: American Mathematical Society, 1992Google Scholar
  17. Z.
    Zwicker, J.: Master’s thesis, McGill University, in preparationGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Aix-Marseille Université, CNRS, CPT, UMR 7332, Case 907MarseilleFrance
  4. 4.Université de Toulon, CNRS, CPT, UMR 7332La GardeFrance
  5. 5.FRUMAMMarseilleFrance

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