齐次度量树上的薛定谔算子：间隙谱
Schrödinger operator on homogeneous metric trees: spectrum in gaps
摘要：
本文研究了齐次根度量树上的Schr dinger算子$A_{gV} = A_0 + gV$的谱性质，该算子具有衰减的实位势$V$和耦合常数$g\ge 0$ .自由拉普拉斯算子$A_0 = -Δ$的谱具有带隙结构，在每个有限间隙的中间具有无限重数的单个本征值。微扰$gV$在带隙中产生额外的本征值。如果势$V$具有固定的符号，则这些特征值是$g$的单调函数。假设后一个条件得到满足，$V$是对称的，即取决于到树根的距离，我们进行了详细的渐近分析的计数函数的离散特征值的极限$g\to\infty$。根据$V$的符号和衰变，这种渐近性要么是外尔型的，要么完全由$V$在无穷远处的行为决定。
英文摘要：
The paper studies the spectral properties of the Schrödinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian $A_0 = -Δ$ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation $gV$ gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed sign. Assuming that the latter condition is satisfied and that $V$ is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$, this asymptotics is either of the Weyl type or is completely determined by the behaviour of $V$ at infinity.
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