(Sub)critical Operators and Spectral Capacities of Rational Frequency Approximants
DOI:
https://doi.org/10.15407/mag20.04.06Анотація
Ми розглядаємо одно-частотні квазіперіодичні оператори Шредингера з аналітичними потенціалами. Позначивши через $S_+$ об'єднання спектрів, взяте за фазами, ми вивчаємо неперервність логарифмічної ємності відносно раціональних наближень. Ми доводимо, що якщо показник Ляпунова є нульовим на спектрі, то ємність спектра для ірраціональної частоти наближувана ємностями спектра для її раціональних наближень.
Mathematical Subject Classification 2020: 30C85, 31A15, 34L40, 47B36
Ключові слова:
логарифмiчна ємнiсть, квазiперiодичнi оператори Шредiнгера, оператор майже Матьє, показник ЛяпуноваПосилання
G. Alpan, A. Goncharov, and B. Hatinoğlu, Some asymptotics for extremal polynomials, Springer Proc. Math. Stat., 155, Computational Analysis, Springer, Cham, 2016, 87--101. https://doi.org/10.1007/978-3-319-28443-9_7
S. Aubry, Metal-insulator transition in one-dimensional deformable lattices, Bifurcation Phenomena in Mathematical Physics and Related Topics, Springer, 1980, 163--184. https://doi.org/10.1007/978-94-009-9004-3_10
A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math. 215 (2015), No. 1, 1--54. https://doi.org/10.1007/s11511-015-0128-7
J. Avron, P.H.M. van Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys. 132 (1990), No. 1, 103--118. https://doi.org/10.1007/BF02278001
J. Avron and B. Simon, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), No. 1, 369--391. https://doi.org/10.1215/S0012-7094-83-05016-0
J. Béllissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. Anal. 48 (1982), No. 3, 408--419. https://doi.org/10.1016/0022-1236(82)90094-5
J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys. 108 (2002), 1203--1218 (Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays). https://doi.org/10.1023/A:1019751801035
W.G. Chambers, Linear-network model for magnetic breakdown in two dimensions, Phys. Rev. 140 (1965), A135--A143. https://doi.org/10.1103/PhysRev.140.A135
J.S. Christiansen, B. Simon, and M. Zinchenko, Asymptotics of Chebyshev polynomials, I: subsets of $R$, Invent. Math. 208 (2017), No. 1, 217--245. https://doi.org/10.1007/s00222-016-0689-x
J.S. Christiansen, B. Simon, and M. Zinchenko, Widom Factors and Szegö-Widom Asymptotics, a Review, Oper. Theory Adv. Appl., 289, Toeplitz operators and random matrices — in memory of Harold Widom, Birkhäuser/Springer, Cham, 2022, 301--319. https://doi.org/10.1007/978-3-031-13851-5_14
H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-77522-5
D. Damanik and J. Fillman, One-dimensional ergodic Schrödinger operators — I. General theory, Graduate Studies in Mathematics, 221, Amer. Math. Soc., Providence, RI, 2022. https://doi.org/10.1090/gsm/221
A. Goncharov and B. Hatinoğlu, Widom factors, Potential Anal. 42 (2015), No. 3, 671--680. https://doi.org/10.1007/s11118-014-9452-3
S. Jitomirskaya and C.A. Marx, Analytic quasi-periodic Schrödinger operators and rational frequency approximants, Geom. Funct. Anal. 22 (2012), No. 5, 1407--1443. https://doi.org/10.1007/s00039-012-0179-2
S. Kalmykov and L.V. Kovalev, Continuity of logarithmic capacity, J. Math. Anal. Appl. 505 (2022), No. 1, Paper No. 125585, 11 pp. https://doi.org/10.1016/j.jmaa.2021.125585
P. Kuchment, An overview of periodic elliptic operators, Bull. Amer. Math. Soc. (N.S.) 53 (2016), No. 3, 343--414. https://doi.org/10.1090/bull/1528
F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), No. 2, 180--195. https://doi.org/10.1006/jath.2001.3571
T. Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.
B. Simon, Almost periodic Schrödinger operators: a review, Adv. in Appl. Math. 3 (1982), No. 4, 463--490. https://doi.org/10.1016/S0196-8858(82)80018-3
B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007), No. 4, 713--772. https://doi.org/10.3934/ipi.2007.1.713
G. Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000. https://doi.org/10.1090/surv/072
V. Totik, The norm of minimal polynomials on several intervals, J. Approx. Theory 163 (2011), No. 6, 738--746. https://doi.org/10.1016/j.jat.2010.07.002
