论文标题
量子步行的光谱和散射特性在同质树上
Spectral and scattering properties of quantum walks on homogenous trees of odd degree
论文作者
论文摘要
For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and三重$（u，u_0，j）$的波算子的完整性。作为应用程序，我们确定一类各向异性量子行走的光谱和散射特性，在具有进化操作员$ u $的奇数树上行走。特别是，我们建立了$ U $的Mourre估算值，获得一类本地$ u $ smooth操作员，并证明$ u $的频谱覆盖了整个单元圆圈，并且纯粹是连续的，在外部可能是有限的套装，其中$ u $可能具有有限多重的特征值。我们还表明，（至少）（至少）三种不同的免费进化运算符选择$ u_0 $是可以证明波动运算符的存在和完整性的三种选择。
For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.
