表面Maryland模型的光谱和传输特性
On the Spectral and Propagation Properties of the Surface Maryland Model
摘要：
我们研究了$\ZZ^d$中的离散薛定谔算子$H$，其表面势形式为$V(x)=g δ(x_1) \tan π(α\cdot x_2+ ω)$，其中对于$x \in \ZZ^d$，我们写为$x=(x_1,x_2), \quad x_1 \in \ZZ^{d_1}, x_2 \in \mathbb{Z}^{d_2}, α\in \R^{d_2}, ω\in [0,1)$。我们首先考虑矢量$α$的分量有理独立的情况，即准周期势的情况。证明了$H$在区间$[-d,d]$上的谱（与离散拉普拉斯算子的谱一致）是绝对连续的.然后，我们表明，广义本征函数对应于这个时间间隔的体积（体）波的形式，这是振荡和非减少（或缓慢减少）的所有变量。它们是入射平面波和由“平面”$\ZZ^{d_2}$散射的无限多个反射或透射平面波的总和。这些本征函数是正交的，完整的，并验证了李普曼-施温格方程的自然模拟。我们还讨论了$d_1=d_2=1$的有理向量$α$的情况，即周期表面势。在这种情况下，我们表明，频谱是绝对连续的，除了体积（布洛赫）波也有表面波，其振幅呈指数衰减为$|x_1| \to \infty$。对应于表面态的光谱部分由有限数量的能带组成。对于大的$q$，在$[-d,d]$之外的谱带在$q$中是指数小的，并且在自然意义上收敛到纯点谱，这在丢番图$α$的情况下在[KP]中发现。
英文摘要：
We study the discrete Schrödinger operator $H$ in $\ZZ^d$ with the surface potential of the form $V(x)=g δ(x_1) \tan π(α\cdot x_2+ ω)$, where for $x \in \ZZ^d$ we write $x=(x_1,x_2), \quad x_1 \in \ZZ^{d_1}, x_2 \in \mathbb{Z}^{d_2}, α\in \R^{d_2}, ω\in [0,1)$. We first consider the case where the components of the vector $α$ are rationally independent, i.e. the case of the quasi periodic potential. We prove that the spectrum of $H$ on the interval $[-d,d]$ (coinciding with the spectrum of the discrete Laplacian) is absolutely continuous. Then we show that generalized eigenfunctions corresponding to this interval have the form of volume (bulk) waves, which are oscillating and non decreasing (or slow decreasing) in all variables. They are the sum of the incident plane wave and of an infinite number of reflected or transmitted plane waves scattered by the "plane" $\ZZ^{d_2}$. These eigenfunctions are orthogonal, complete and verify a natural analogue of the Lippmann-Schwinger equation. We also discuss the case of rational vectors $α$ for $d_1=d_2=1$, i.e. a periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides volume (Bloch) waves there are also surface waves, whose amplitude decays exponentially as $|x_1| \to \infty$. The part of the spectrum corresponding to the surface states consists of a finite number of bands. For large $q$ the bands outside of $[-d,d]$ are exponentially small in $q$, and converge in a natural sense to the pure point spectrum, that was found in [KP] in the case of the Diophantine $α$'s.
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