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This article is cited in 57 scientific papers (total in 57 papers)
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.
Received: 28.06.2006 Revised: 18.07.2006
Citation:
A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces”, Mat. Zametki, 80:6 (2006), 864–884; Math. Notes, 80:6 (2006), 814–832
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https://www.mathnet.ru/eng/mzm3363https://doi.org/10.4213/mzm3363 https://www.mathnet.ru/eng/mzm/v80/i6/p864
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