On the eigenvalues of the Sturm--Liouville operator with potentials from Sobolev spaces

A. M. Savchuk ; A. A. Shkalikov

Matematičeskie zametki, Tome 80 (2006) no. 6, pp. 864-884

Résumé

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.