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On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces
A. M. Savchuk,
A. A. Shkalikov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We study the asymptotic behavior of the eigenvalues the Sturm–Liouville operator
$Ly= -y'' +q(x)y$ with potentials from the Sobolev space
$W_2^{\theta-1}$,
$\theta\ge0$, including the nonclassical case
$\theta\in[0,1)$ in which the potential is a distribution. The results are obtained in new terms. Let
$s_{2k}(q)=\lambda_{k}^{1/2}(q)-k$,
$s_{2k-1}(q)=\mu_{k}^{1/2}(q)-k-1/2$, where
$\{\lambda_k\}_1^{\infty}$ and
$\{\mu_k\}_1^{\infty}$ are the sequences of eigenvalues of the operator
$L$ generated by the Dirichlet and Dirichlet–Neumann boundary conditions, respectively. We construct special Hilbert spaces
$\hat\ell_2^{\,\theta}$ such that the mapping
$F\colon W^{\theta-1}_2\to\hat\ell_2^{\,\theta}$ defined by
the equality
$F(q)=\{s_n\}_1^{\infty}$ is well defined for all
$\theta\ge0$. The main result is as follows: for
$\theta>0$, the mapping
$F$ is weakly nonlinear, i.e., can be expressed as
$F(q)=Uq+\Phi(q)$, where
$U$ is the isomorphism of the spaces
$W^{\theta-1}_2$ and
$\hat\ell_2^{\,\theta}$, and
$\Phi(q)$ is a compact mapping. Moreover, we prove the estimate
$\|\Phi(q)\|_{\tau}\le C\|q\|_{\theta-1}$, where the exact value of
$\tau=\tau(\theta)>\theta-1$ is given and the constant
$C$ depends only on the radius of the ball
$\|q\|_{\theta-1}\le R$, but is independent of the function
$q$ varying in this ball.
UDC:
517.984 Received: 28.06.2006
Revised: 18.07.2006
DOI:
10.4213/mzm3363
Fulltext:
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