Letτbe a generalized Thue-Morse substitution on a two-letter alphabet{a,b}:τ(a)=ambm,τ(b)=bmam for the integer m≥2.Letξbe a sequence in{a,b}Z that is generated byτ.We study the one-dimensional Schr?dinger operat...Letτbe a generalized Thue-Morse substitution on a two-letter alphabet{a,b}:τ(a)=ambm,τ(b)=bmam for the integer m≥2.Letξbe a sequence in{a,b}Z that is generated byτ.We study the one-dimensional Schr?dinger operator Hm,λon l2(Z)with a potential given by v(n)=λVξ(n),whereλ>0 is the coupling and Vξ(n)=1(Vξ(n)=-1)ifξ(n)=a(ξ(n)=b).LetΛ2=2,and for m>2,letΛm=m if m≡0 mod 4;letΛm=m-3 if m≡1 mod 4;letΛm=m-2if m≡2 mod 4;letΛm=m-1 if m≡3 mod 4.We show that the Hausdorff dimension of the spectrumσ(Hm,λ)satisfies that dimHσ(Hm,λ)>logΛm/(log 64m+4).It is interesting to see that dimHσσ(Hm,λ)tends to 1 as m tends to infinity.展开更多
众所周知,在光子带隙材料/结构中电磁波存在两种传播模式。一种模式是,在不考虑材料/结构的吸收效应情况下,电磁波不随传播距离增大而衰减,人们称这种电磁波为正常传播模式的电磁波（简记为OPM EM waves,Ordinary-Propagation-Mode elec...众所周知,在光子带隙材料/结构中电磁波存在两种传播模式。一种模式是,在不考虑材料/结构的吸收效应情况下,电磁波不随传播距离增大而衰减,人们称这种电磁波为正常传播模式的电磁波（简记为OPM EM waves,Ordinary-Propagation-Mode electromagnetic waves）。另一种模式是,电磁波随传播距离增大而衰减,人们称这种电磁波为衰减传播模式的电磁。展开更多
By means of the theory of electromagnetic wave propagation and transfer matrix method, this paper investigates the band rules for the frequency spectra of three kinds of one-dimensional （1D） aperiodic photonic cryst...By means of the theory of electromagnetic wave propagation and transfer matrix method, this paper investigates the band rules for the frequency spectra of three kinds of one-dimensional （1D） aperiodic photonic crystals （PCs）, generalized Fibonacci GF（p, 1）, GF（1,2）, and Thue Morse （TM） PCs, with negative refractive index （NRI） materials. It is found that all of these PCs can open a broad zero-n gap, TM PC possesses the largest zero-n gap, and with the increase of p, the width of the zero-n gap for GF（p, 1） PC becomes smaller. This characteristic is caused by the symmetry of the system and the open position of the zero-n gap. It is found that for GF（p, 1） PCs, the possible limit zero-n gaps open at lower frequencies with the increase of p, but for GF（1,2） and TM PCs, their limit zero-n gaps open at the same frequency. Additionally, for the tbree bottom-bands, we find the interesting perfect self-similarities of the evolution structures with the increase of generation, and obtain the corresponding subband-number formulae. Based on 11 types of evolving manners Qi （i = 1, 2,... , 11） one can plot out the detailed evolution structures of the three kinds of aperiodic PCs for any generation.展开更多
The m-tuplings Morse sequence, as a fixed point of a constant length substitution, was discussed. An equivalent definition was given by the property of the m-adie development of the integers. Using the combinatorial p...The m-tuplings Morse sequence, as a fixed point of a constant length substitution, was discussed. An equivalent definition was given by the property of the m-adie development of the integers. Using the combinatorial properties of the m-tuplings Morse sequence, we mainly obtain this sequenee is an admissible sequence and the other two sequenees generated by it are also admissible sequences, which extends the conclusion that the Thue-Morse sequence is an admissible sequences.展开更多
基金supported by the National Natural ScienceFoundation of China(11871098)。
文摘Letτbe a generalized Thue-Morse substitution on a two-letter alphabet{a,b}:τ(a)=ambm,τ(b)=bmam for the integer m≥2.Letξbe a sequence in{a,b}Z that is generated byτ.We study the one-dimensional Schr?dinger operator Hm,λon l2(Z)with a potential given by v(n)=λVξ(n),whereλ>0 is the coupling and Vξ(n)=1(Vξ(n)=-1)ifξ(n)=a(ξ(n)=b).LetΛ2=2,and for m>2,letΛm=m if m≡0 mod 4;letΛm=m-3 if m≡1 mod 4;letΛm=m-2if m≡2 mod 4;letΛm=m-1 if m≡3 mod 4.We show that the Hausdorff dimension of the spectrumσ(Hm,λ)satisfies that dimHσ(Hm,λ)>logΛm/(log 64m+4).It is interesting to see that dimHσσ(Hm,λ)tends to 1 as m tends to infinity.
文摘众所周知,在光子带隙材料/结构中电磁波存在两种传播模式。一种模式是,在不考虑材料/结构的吸收效应情况下,电磁波不随传播距离增大而衰减,人们称这种电磁波为正常传播模式的电磁波（简记为OPM EM waves,Ordinary-Propagation-Mode electromagnetic waves）。另一种模式是,电磁波随传播距离增大而衰减,人们称这种电磁波为衰减传播模式的电磁。
基金Project supported by the National Natural Science Foundation of China (Grant No 10974061)the Program for Innovative Research Team of the Higher Education of Guangdong Province of China (Grant No 06CXTD005)
文摘By means of the theory of electromagnetic wave propagation and transfer matrix method, this paper investigates the band rules for the frequency spectra of three kinds of one-dimensional （1D） aperiodic photonic crystals （PCs）, generalized Fibonacci GF（p, 1）, GF（1,2）, and Thue Morse （TM） PCs, with negative refractive index （NRI） materials. It is found that all of these PCs can open a broad zero-n gap, TM PC possesses the largest zero-n gap, and with the increase of p, the width of the zero-n gap for GF（p, 1） PC becomes smaller. This characteristic is caused by the symmetry of the system and the open position of the zero-n gap. It is found that for GF（p, 1） PCs, the possible limit zero-n gaps open at lower frequencies with the increase of p, but for GF（1,2） and TM PCs, their limit zero-n gaps open at the same frequency. Additionally, for the tbree bottom-bands, we find the interesting perfect self-similarities of the evolution structures with the increase of generation, and obtain the corresponding subband-number formulae. Based on 11 types of evolving manners Qi （i = 1, 2,... , 11） one can plot out the detailed evolution structures of the three kinds of aperiodic PCs for any generation.
基金Supported by the Special Funds for Major StateBasic Research Projects (60472041)
文摘The m-tuplings Morse sequence, as a fixed point of a constant length substitution, was discussed. An equivalent definition was given by the property of the m-adie development of the integers. Using the combinatorial properties of the m-tuplings Morse sequence, we mainly obtain this sequenee is an admissible sequence and the other two sequenees generated by it are also admissible sequences, which extends the conclusion that the Thue-Morse sequence is an admissible sequences.
