On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces
On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces
复制标题
DOI：
10.1007/s11006-006-0204-6
复制
发表时间：
2006-09
影响因子：
0.6
通讯作者：
A. Savchuk;A. Shkalikov
中科院分区：
文献类型：
--
作者：
A. Savchuk;A. Shkalikov
研究了Sobolev空间w2θ−1，θ≥0的势的Sturm-Liouville算子的特征值= - y″+q(x)y的渐近行为，包括势为分布的非经典情况θ∈[0,1]。结果是用新的术语得出的。Lets2k(q) =λk1/2(q)−k,s2k−1(q) =μk1/2(q)−k−1/2，其中{λk}1∞和{μk}1∞分别是由Dirichlet和Dirichlet- neumann边界条件生成的算子l的特征值序列。我们构造了特殊的希尔伯特空间2θ，使得由等式f (q) = {sn}1∞定义的映射f:W2θ−1→2θ对于所有θ≥0是定义好的。主要结果如下：对于θ>，映射f_1是弱非线性的，即可以表示为f (q) =Uq+ Φ(q)，其中u是空间w2θ−1和t2θ的同构，Φ(q)是紧映射。此外，我们证明了估计∥Ф(q)∥τ≤C∥q∥θ−1，其中给出了τ=τ(θ) >θ−1的精确值，并且常数只依赖于球的半径∥q∥θ−≤R，而与球中的函数无关。
We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operatorLy= −y″ +q(x)ywith potentials from the Sobolev spaceW2θ−1,θ≥ 0, including the nonclassical caseθ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Lets2k(q) =λk1/2(q) −k,s2k−1(q) =μk1/2(q) −k− 1/2, where {λk}1∞and {μk}1∞are the sequences of eigenvalues of the operatorLgenerated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spacest2θsuch that the mappingF:W2θ−1→t2θdefined by the equalityF(q) = {sn}1∞is well defined for allθ≥ 0. The main result is as follows: forθ> 0, the mappingFis weakly nonlinear, i.e., can be expressed asF(q) =Uq+ Φ(q), whereUis the isomorphism of the spacesW2θ−1andt2θ, and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥τ≤C∥q∥θ−1, where the exact value ofτ=τ(θ) >θ− 1 is given and the constantCdepends only on the radius of the ball ∥q∥θ−≤R, but is independent of the functionqvarying in this ball.