具有索伯列夫空间位势的Sturm-Liouville算子的特征值
On the eigenvalues of Sturm--Liouville operators with potentials from Sobolev spaces
摘要：
本文研究了Strum-Liouville算子$Ly= -y'' +q(x)y $的特征值的渐近性质，其中位势来自于索伯列夫空间$W_2^{θ-1}, θ\geqslant 0$，包括非经典情形下的$θ\in [0,1)$.结果是在新的条件下得到的。定义数$$ s_{2k}(q)= λ_{k}^{1/2}(q)-k, \quad s_{2k-1}(q)= μ_{k}^{1/2}(q)-k-1/2, $$，其中$\{λ_k\}_1^{\infty}$和$\{μ_k\}_1^{\infty}$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Hilbert空间$\hat l_2^θ$，使得由公式$F(q)=\{s_n\}_1^{\infty}$定义的映射$F: W^{θ-1}_2 \to \hat l_2^θ$对所有的$θ\geqslant 0$都是良好定义的.主要结果如下：对所有固定的$θ>0$，映射$F$是弱非线性的，即它有一个形式为$F(q) =Uq+Φ(q)$的表示，其中$U$是空间$W^{θ-1}_2 $和$\hat l_2^θ$之间的同构，$Φ(q)$是紧映射.此外，我们证明了估计$\|Φ(q)\|_τ \leqslant C\|q\|_{θ-1}$，其中$τ=τ(θ)>θ$的值是明确给出的，常数$C$只依赖于球的半径$\|q\|_θ \leqslant R$，但不依赖于函数$q$，通过这个球
英文摘要：
We study asymptotic behavior of the eigenvalues of Strum--Liouville operators $Ly= -y'' +q(x)y $ with potentials from Sobolev spaces $W_2^{θ-1}, θ\geqslant 0$, including the non-classical case $θ\in [0,1)$ when the potentials are distributions. The results are obtained in new terms. Define the numbers $$ s_{2k}(q)= λ_{k}^{1/2}(q)-k, \quad s_{2k-1}(q)= μ_{k}^{1/2}(q)-k-1/2, $$ where $\{λ_k\}_1^{\infty}$ and $\{μ_k\}_1^{\infty}$ are the sequences of the eigenvalues of the operator $L$ generated by the Dirichlet and Dirichlet--Neumann boundary conditions, respectivaly. We construct special Hilbert spaces $\hat l_2^θ$ such that the map $F: W^{θ-1}_2 \to \hat l_2^θ$, defined by formula $F(q)=\{s_n\}_1^{\infty}$, is well-defined for all $θ\geqslant 0$. The main result is the following: for all fixed $θ>0$ the map $F$ is weekly nonlinear, i.e. it admits a representation of the form $F(q) =Uq+Φ(q)$, where $U$ is the isomorphism between the spaces $W^{θ-1}_2 $ and $\hat l_2^θ$, and $Φ(q)$ is a compact map. Moreover we prove the estimate $\|Φ(q)\|_τ \leqslant C\|q\|_{θ-1}$, where the value of $τ=τ(θ)>θ$ is given explicitly and the constant $C$ depends only of the radius of the ball $\|q\|_θ \leqslant R$ but does not depend on the function $q$, running through this ball
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