SPECTRAL PROPERTIES OF ONE-DIMENSIONAL SCHRODINGER-OPERATORS WITH POTENTIALS GENERATED BY SUBSTITUTIONS

被引:92
作者
BOVIER, A
GHEZ, JM
机构
[1] CTR PHYS THEOR,F-13288 MARSEILLE 9,FRANCE
[2] UNIV TOULON & VAR,DEPT MATH,PHYMAT,F-83957 LA GARDE,FRANCE
关键词
D O I
10.1007/BF02097231
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We investigate one-dimensional discrete Schrodinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schrodinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.
引用
收藏
页码:45 / 66
页数:22
相关论文
共 33 条
[21]   CANTOR SPECTRA AND SCALING OF GAP WIDTHS IN DETERMINISTIC APERIODIC SYSTEMS [J].
LUCK, JM .
PHYSICAL REVIEW B, 1989, 39 (09) :5834-5849
[22]  
MARTINELLI F, 1987, RIV NUOVO CIMENTO, V10
[23]  
Morse M., 1921, T AM MATH SOC, V22, P84
[24]   ONE-DIMENSIONAL SCHRODINGER-EQUATION WITH AN ALMOST PERIODIC POTENTIAL [J].
OSTLUND, S ;
PANDIT, R ;
RAND, D ;
SCHELLNHUBER, HJ ;
SIGGIA, ED .
PHYSICAL REVIEW LETTERS, 1983, 50 (23) :1873-1876
[25]  
QUEFFELEC M, 1987, LECTURE NOTES MATH, V1294
[26]  
Reed M., 1972, METHODS MODERN MATH
[27]  
Riklund R., 1987, International Journal of Modern Physics B, V1, P121, DOI 10.1142/S0217979287000104
[28]  
SHAPIRO HS, 1959, P AM MATH SOC, V10, P855
[29]   METALLIC PHASE WITH LONG-RANGE ORIENTATIONAL ORDER AND NO TRANSLATIONAL SYMMETRY [J].
SHECHTMAN, D ;
BLECH, I ;
GRATIAS, D ;
CAHN, JW .
PHYSICAL REVIEW LETTERS, 1984, 53 (20) :1951-1953
[30]  
STEINHARDT PJ, 1992, THESIS GRONINGEN