Research PaperNo Access

Transport exponents of states with large support

Vitalii Gerbuz

Mathematics Department, Rice University, Houston, TX77251–1892, USA

https://doi.org/10.1142/S0129055X19500296Cited by:0 (Source: Crossref)

Abstract

We investigate spreading rates of one-dimensional quantum states under the Schrödinger time-evolution. The focus of this paper is on the states that either have finite support or decay exponentially at $\pm \infty$ $\pm \infty$ . In particular, we extend results of Damanik and Tcheremchantsev on estimating transport exponents that were originally proved to hold for the initial states supported on a single site. These general upper and lower estimates are then applied to several classes of models, including Sturmian, quasi-periodic and substitution-generated potentials, and the random polymer model.

Keywords:

AMSC: 81Q10, 34K08

References

1. J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys. 108(5) (2002) 1203–1218. Web of Science, Google Scholar
2. S. Cantat, Bers and Hénon, Painlevé and Schrödinger, Duke Math. J. 149(3) (2009) 411–460. Web of Science, Google Scholar
3. J.-M. Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, 192 (1993) 59–68. Google Scholar
4. D. Damanik and S. Tcheremchantsev, A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators, Discrete Contin. Dyn. Syst. A 28 (2010) 1381–1412. Web of Science, Google Scholar
5. D. Damanik, A. Gorodetski, Q. Liu and Y. Qu, Transport exponents of Sturmian Hamiltonians, J. Funct. Anal. 269(5) (2015) 1404–1440. Web of Science, Google Scholar
6. D. Damanik, A. Gorodetski and W. Yessen, The Fibonacci Hamiltonian, Invent. Math. 206(3) (2016) 629–692. Web of Science, ADS, Google Scholar
7. D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, III. $α$ $α$ -continuity, Comm. Math. Phys. 212(1) (2000) 191–204. Web of Science, ADS, Google Scholar
8. D. Damanik, D. Lenz and G. Stolz, Lower transport bounds for one-dimensional continuum Schrödinger operators, Math. Ann. 336(2) (2006) 361–389. Web of Science, Google Scholar
9. D. Damanik and S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Comm. Math. Phys. 236(3) (2003) 513–534. Web of Science, ADS, Google Scholar
10. D. Damanik and S. Tcheremchantsev, Upper bounds in quantum dynamics, J. Amer. Math. Soc. 20 (2007) 799–827. Web of Science, Google Scholar
11. D. Damanik and S. Tcheremchantsev, Quantum dynamics via complex analysis methods: General upper bounds without time-averaging and tight lower bounds for the strongly coupled Fibonacci Hamiltonian, J. Funct. Anal. 255(10) (2008) 2872–2887. Web of Science, Google Scholar
12. F. Germinet, A. Kiselev and S. Tcheremchantsev, Transfer matrices and transport for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 54(3) (2004) 787–830. Web of Science, Google Scholar
13. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10(2) (1989) 95–100. ADS, Google Scholar
14. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra I. Half-line operators, Acta Math. 183(2) (1999) 171–189. Web of Science, Google Scholar
15. S. Jitomirskaya and R. Mavi, Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials, Int. Math. Res. Not. 2017 (2017) 96–120. Web of Science, Google Scholar
16. S. Jitomirskaya and H. Schulz-Baldes, Upper bounds on wavepacket spreading for random Jacobi matrices, Comm. Math. Phys. 273(3) (2007) 601–618. Web of Science, ADS, Google Scholar
17. S. Jitomirskaya, H. Schulz-Baldes and G. Stolz, Delocalization in random polymer models, Comm. Math. Phys. 233 (2003) 27–48. Web of Science, ADS, Google Scholar
18. L. Yoram, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142(2) (1996) 406–445. Web of Science, Google Scholar
19. M. Reed and B. Simon, Scattering Theory. Methods of Modern Mathematical Physics: III. Scattering Theory (Academic Press, 1979). Google Scholar
20. E. Sorets and T. Spencer, Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys. 142(3) (1991) 543–566. Web of Science, ADS, Google Scholar
21. A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111(3) (1987) 409–415. Web of Science, ADS, Google Scholar
22. G. Teschl, Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators (American Mathematical Society, 2009). Google Scholar