This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all...This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.展开更多
In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(Rs). Suppose ψ = (ψ1, . . . , ψr)T and ψ = ( ψ1, . . . , ψr)T are two compactly supported vector...In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(Rs). Suppose ψ = (ψ1, . . . , ψr)T and ψ = ( ψ1, . . . , ψr)T are two compactly supported vectors of functions in the Sobolev space (Hμ(Rs))r for some μ > 0. We provide a characterization for the sequences {ψjk : = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jk : = 1, . . . , r, j ∈ Z, k ∈ Zs} to form two Riesz sequences for L2(Rs), where ψjk = mj/2ψ (M j ·k) and ψjk = mj/2 ψ (M j ·k), M is an s × s integer matrix such that limn→∞ Mn = 0 and m = |detM|. Furthermore, let = (1, . . . , r)T and = ( 1, . . . , r)T be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, a and M, where a and a are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1, . . . , ψνr)T and ψν = ( ψν1, . . . , ψ?νr)T , ν = 1, . . . , m 1 such that two sequences {ψjνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} form two Riesz multiwavelet bases for L2(Rs). The bracket product [f, g] of two vectors of functions f, g in (L2(Rs))r is an indispensable tool for our characterization.展开更多
基金the National Natural Science Foundation of China (Grant No.10562002) the Natural Science Foundation of Inner Mongolia (Grant Nos.200508010103,200711020106)
文摘This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.
基金supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)
文摘In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(Rs). Suppose ψ = (ψ1, . . . , ψr)T and ψ = ( ψ1, . . . , ψr)T are two compactly supported vectors of functions in the Sobolev space (Hμ(Rs))r for some μ > 0. We provide a characterization for the sequences {ψjk : = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jk : = 1, . . . , r, j ∈ Z, k ∈ Zs} to form two Riesz sequences for L2(Rs), where ψjk = mj/2ψ (M j ·k) and ψjk = mj/2 ψ (M j ·k), M is an s × s integer matrix such that limn→∞ Mn = 0 and m = |detM|. Furthermore, let = (1, . . . , r)T and = ( 1, . . . , r)T be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, a and M, where a and a are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1, . . . , ψνr)T and ψν = ( ψν1, . . . , ψ?νr)T , ν = 1, . . . , m 1 such that two sequences {ψjνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} form two Riesz multiwavelet bases for L2(Rs). The bracket product [f, g] of two vectors of functions f, g in (L2(Rs))r is an indispensable tool for our characterization.
