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Spectral and dynamical properties of sparse one-dimensional continuous Schrödinger and Dirac operators

Part of: Spectral theory and eigenvalue problems General mathematical topics and methods in quantum theory Elliptic equations and systems

Published online by Cambridge University Press:  28 June 2013

Silas L. Carvalho  and
César R. de Oliveira
Silas L. Carvalho
Affiliation:
Instituto de Ciência e Technologia – UNIFESP, São José dos Campos, SP 12231–280, Brazil ([email protected])
César R. de Oliveira
Affiliation:
Departamento de Matemática – UFSCar, São Carlos, SP 13560–970, Brazil ([email protected])
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Abstract

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Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.

Keywords

spectral theorysparse potentialSchrödinger operatorsDirac operators

MSC classification

Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 35P05: General topics in linear spectral theory 35J10: Schrödinger operator
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 667 - 700
Copyright
Copyright © Edinburgh Mathematical Society 2013 

References

1.Carvalho, S. L., de Oliveira, C. R. and Prado, R. A., Sparse one-dimensional discrete Dirac operators, II, spectral properties, J. Math. Phys. 52 (2011), 073501.CrossRefGoogle Scholar
2.Carvalho, S. L., Marchetti, D. H. U. and Wreszinski, W. F., Sparse Block–Jacobi matrices with accurate Hausdorff dimension, J. Math. Analysis Applic. 368 (2010), 218–234.CrossRefGoogle Scholar
3.Carvalho, S. L., Marchetti, D. H. U. and Wreszinski, W. F., On the uniform distribution of the Prüfer angles and its implication to a sharp transition of Jacobi matrices with randomly sparse perturbations, eprint (arXiv:1006.2849, 2011).Google Scholar
4.Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (Krieger, New York, 1984).Google Scholar
5.Combes, J. M. and Mantica, G., Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, in Long time behaviour of classical and quantum systems (ed. S. Graffi and A. Martinez), Series on Concrete and Applied Mathematics, Volume 1, pp. 107–123 (World Scientific, 2001).CrossRefGoogle Scholar
6.Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B., Schrödinger operators with application to quantum mechanics and global geometry (Springer, 1987).Google Scholar
7.Damanik, D., Lenz, D. and Stolz, G., Lower transport bounds for one-dimensional continuum Schrödinger operators, Math. Annalen 336 (2006), 361–369.CrossRefGoogle Scholar
8.de Oliveira, C. R. and Prado, R. A., Dynamical localization for the 1D Bernoulli discrete Dirac operator, J. Phys. A 38 (2005), L115–L119.CrossRefGoogle Scholar
9.de Oliveira, C. R. and Prado, R. A., Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator, J. Math. Phys. 46 (2005), 072105.CrossRefGoogle Scholar
10.Falconer, K. J., Fractal geometry (Wiley, 1990).Google Scholar
11.Germinet, F., Kiselev, A. and Tcheremchantsev, S., Transfer matrices and transport for Schrödinger operators, Annales Inst. Fourier 54 (2004), 787–830.CrossRefGoogle Scholar
12.Gilbert, D. J. and Pearson, D. B., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Analysis Applic. 128 (1987), 30–56.CrossRefGoogle Scholar
13.Hundertmark, D. and Kirsch, W., Spectral theory of sparse potentials, in Stochastic processes, physics and geometry: new interplays: a volume in honor of Sergio Albeverio, Canadian Mathematical Society Conference Proceedings Series, Volume 28, pp. 213–238 (American Mathematical Society, Providence, RI, 2000).Google Scholar
14.Jitomirskaya, S. and Last, Y., Power-law subordinacy and singular spectra, I, half-line operators, Acta Math. 183 (1999), 171–189.CrossRefGoogle Scholar
15.Kiselev, A., Last, Y. and Simon, B., Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Commun. Math. Phys. 194 (1998), 1–45.CrossRefGoogle Scholar
16.Krutikov, D., Asymptotics of the Fourier transform of the spectral measure for Schrödinger operators with bounded and unbounded sparse potentials, J. Phys. A 35 (2002), 6393–6417.Google Scholar
17.Kuipers, L. and Niederreiter, H., Uniform distributions of sequences (Wiley, 1974).Google Scholar
18.Last, Y., Quantum dynamics and decomposition of singular continuous spectra, J. Funct. Analysis 142 (1996), 406–445.CrossRefGoogle Scholar
19.Last, Y., Sturm–Liouville operator on infinite intervals, in Sturm–Liouville theory: past and present (ed. W. O. Amrein, A. M. Hinz and D. B. Pearson), pp. 99–120 (Birkhäuser, 2005).CrossRefGoogle Scholar
20.Last, Y. and Simon, B., Eigenfunctions, transfer matrices and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329–367.CrossRefGoogle Scholar
21.Levitan, B. M. and Sargsjan, I. S., Introduction to spectral theory (American Mathematical Society, Providence, RI, 1975).Google Scholar
22.Marchetti, D. H. U., Wreszinski, W. F., Guidi, L. F. and Angelo, R. M., Spectral transition in a sparse model and a class of nonlinear dynamical systems, Nonlinearity 20 (2007), 765–787.CrossRefGoogle Scholar
23.Pearson, D. B., Singular continuous measures in the scattering theory, Commun. Math. Phys. 60 (1978), 13–36.CrossRefGoogle Scholar
24.Prado, R. A. and de Oliveira, C. R., Sparse 1D discrete Dirac operators, I, quantum transport, J. Math. Analysis Applic. 385 (2012), 947–960.CrossRefGoogle Scholar
25.Rogers, C. A., Hausdorff measures, 2nd edn (Cambridge University Press, 1998).Google Scholar
26.Rogers, C. A. and Taylor, S. J., The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273–302.CrossRefGoogle Scholar
27.Simon, B., Spectral analysis of rank one perturbations and applications, in Mathematical quantum theory II: Schrödinger operators (ed. J. Feldman, R. Froese and L. Rosen), CRM Proceedings and Lecture Notes, Volume 8, pp. 109–149 (American Mathematical Society, Providence, RI, 1995).Google Scholar
28.Simon, B. and Stolz, G., Operators with singular continuous spectrum, V, sparse potentials, Proc. Am. Math. Soc. 124 (1996), 2073–2080.CrossRefGoogle Scholar
29.Tcheremchantsev, S., Dynamical analysis of Schrödinger operators with growing sparse potentials, Commun. Math. Phys. 253 (2005), 221–252.CrossRefGoogle Scholar
30.Teschl, H., Jacobi operators and completely integrable nonlinear lattices (American Mathematical Society, Providence, RI, 2000).Google Scholar
31.Thaller, B., The Dirac equation (Springer, 1992).CrossRefGoogle Scholar
32.Wilcox, C. H., Scattering states and wave operators in the abstract theory of scattering, J. Funct. Analysis 12 (1973), 257–274.CrossRefGoogle Scholar
33.Zlatoš, A., Sparse potentials with fractional Hausdorff dimension, J. Funct. Analysis 207 (2004), 216–252.CrossRefGoogle Scholar