论文标题
关于希尔伯特变换操作员在多间隙的光谱特性上
On the spectral properties of the Hilbert transform operator on multi-intervals
论文作者
论文摘要
令$ j，e \ subset \ mathbb r $为两个具有非相互作用的内饰的多间隙。考虑以下操作员$$ a：\，l^2（j）\至l^2（e），\（af）（x）= \ frac1π\ int_ {j} \ frac {f（y）\ text {d} y} y} y} y} {x-y} {x-y} {x y}，$ a^\ a^\ dagger $我们介绍了一个自偶有操作员$ \ Mathscr k $作用于$ l^2（e）\ oplus l^2（j）$，其偏高块由$ a $ a $ a $ a $ a a $ a^\ dagger $组成。在本文中，我们研究了$ \ Mathscr k $的光谱属性和运营商$ a^\匕首a $和$ a a^\匕首$。我们的主要工具是使用适当的Riemann-Hilbert问题获得$ \ Mathscr k $的分解，该分解用$ \ Mathscr r $表示，然后计算频谱参数$λ$中的$ \ Mathscr r $的跳跃和杆子。我们表明，$ \ Mathscr k $的频谱具有绝对连续的组件$ [0,1] $，并且仅当$ j $和$ e $具有共同的终点，并且其多重性等于它们的数字。如果没有共同的终点，则$ \ Mathscr k $的光谱仅由特征值和$ 0 $组成。如果有共同的终点，则$ \ mathscr k $可能会在连续频谱中嵌入特征值，它们每个都有有限的多重性，而特征值可能仅在$ 0 $中累积。在所有情况下，$ \ Mathscr k $都不具有奇异的连续频谱。 $ A^\ Dagger的光谱属性也与$ \ Mathscr k $非常相似。
Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $λ$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\dagger A$ and $A A^\dagger$, which are very similar to those of $\mathscr K$, are obtained as well.
