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

Abstract

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