Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree -2
Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree -2
复制标题
具有-2次势渐近齐次性的薛定谔算子的特征值
DOI：
10.1090/s0002-9947-08-04479-6
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发表时间：
2005
影响因子：
1.3
通讯作者：
Simon Marshall
中科院分区：
文献类型：
--
作者：
Andrew Hassell;Simon Marshall
我们加强和推广了Kirsch和Simon关于函数NL（E）的行为的结果，即算子L = A + V在Rd下-E中的束缚态数.这里V是有界势，渐近地表现为P（ω）r-2，其中P是球面上的函数。众所周知，这样的算子的本征值都是非正的，并且只在0处累积。若球面S d-1上的算子Δ S d-1 + P有负特征值-μ1，..... -μ n小于-（d - 2）2 /4，我们证明了NL（E）可以估计为NL（E）=log（E-1）2πn i=1 <$μi-（d-2）2 /4+O（1）.因此，特别地，如果没有这样的负特征值，则L具有有限离散谱。此外，在一些额外的假设下，包括d = 3，且恰好有一个特征值-μ1小于-1/4，且所有其他特征值> -1/4，我们证明了负谱渐近于一个比率为exp（-2π/μ1-1/4）的几何级数.
We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function N L (E), the number of bound states of the operator L = A + V in R d below -E. Here V is a bounded potential behaving asymptotically like P(ω)r -2 where P is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator Δ S d-1 + P on the sphere S d-1 has negative eigenvalues -μ1,....., -μ n less than -(d - 2) 2 /4, we prove that N L (E) may be estimated as N L (E)=log(E -1 ) 2πnΣi=1√μi-(d-2) 2 /4+O(1). Thus, in particular, if there are no such negative eigenvalues, then L has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that d = 3 and that there is exactly one eigenvalue -μ1 less than -1/4, with all others > -1/4, we show that the negative spectrum is asymptotic to a geometric progression with ratio exp(-2π/√μ1-1 4).