SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS
SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS
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齐次度量树上的薛定谔算子：间隙谱
DOI：
10.1142/s0129055x02001235
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发表时间：
2001
影响因子：
1.8
通讯作者：
M. Solomyak
中科院分区：
文献类型：
--
作者：
A. Sobolev;M. Solomyak
研究了齐次根度量树上的Schrodinger算子AgV = A0 + gV的谱性质，其中V是衰减实值势，耦合常数g ≥ 0.自由拉普拉斯算子A0 = -Δ的谱具有带隙结构，在每个有限隙的中间具有无限重数的单个本征值。微扰gV引起间隙中的额外本征值。这些特征值是g的单调函数，如果势V有一个固定的符号。假设满足后一个条件，并且V是对称的，即取决于到树根的距离，我们对极限g → ∞中离散特征值的计数函数进行了详细的渐近分析。根据V的符号和衰减，这个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。
The paper studies the spectral properties of the Schrodinger operator AgV = A0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0. The spectrum of the free Laplacian A0 = -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 → ∞. 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.