一维周期Dirac算子Riesz投影的Bari-Markus性质
Bari-Markus property for Riesz projections of 1D periodic Dirac operators
摘要：
在具有周期、反周期或Dirichlet边界条件$(bc)$的$[0,π]$上考虑的具有$L^2$ -势$$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,π]), $$的Dirac算子$$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π],$$具有离散谱，如果$N $足够大，则对于$|n| \geq N$，Riesz投影$$ S_N = \frac{1}{2πi} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2πi} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$是良好定义的.证明了自由算子的Riesz投影是$$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty, $$，其中$P_n^0, n \in \mathbb{Z},$是自由算子的Riesz投影.然后，利用Bari-Markus准则，证明了谱Riesz分解$$ f = S_N f + \sum_{|n| >N} P_n f, \quad \forall f \in L^2; $$在$L^2.$中无条件收敛
英文摘要：
The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,π]), $$ considered on $[0,π]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$ S_N = \frac{1}{2πi} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2πi} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$ are well--defined for $|n| \geq N$ if $N $ is sufficiently large. It is proved that $$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty, $$ where $P_n^0, n \in \mathbb{Z},$ are the Riesz projections of the free operator. Then, by the Bari--Markus criterion, the spectral Riesz decompositions $$ f = S_N f + \sum_{|n| >N} P_n f, \quad \forall f \in L^2; $$ converge unconditionally in $L^2.$
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