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28 March 2024
 
  » arxiv » math/0609213

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On the eigenvalues of Sturm--Liouville operators with potentials from Sobolev spaces
A.M. Savchuk ; A.A. Shkalikov ;
Date 7 Sep 2006
Subject Functional Analysis; Spectral Theory
AbstractWe study asymptotic behavior of the eigenvalues of Strum--Liouville operators $Ly= -y’’ +q(x)y $ with potentials from Sobolev spaces $W_2^{ heta -1}, heta geqslant 0$, including the non-classical case $ heta in [0,1)$ when the potentials are distributions. The results are obtained in new terms. Define the numbers $$ s_{2k}(q)= lambda_{k}^{1/2}(q)-k, quad s_{2k-1}(q)= mu_{k}^{1/2}(q)-k-1/2, $$ where ${lambda_k}_1^{infty}$ and ${mu_k}_1^{infty}$ are the sequences of the eigenvalues of the operator $L$ generated by the Dirichlet and Dirichlet--Neumann boundary conditions, respectivaly. We construct special Hilbert spaces $hat l_2^{ heta}$ such that the map $F: W^{ heta-1}_2 o hat l_2^{ heta}$, defined by formula $F(q)={s_n}_1^{infty}$, is well-defined for all $ hetageqslant 0$. The main result is the following: for all fixed $ heta>0$ the map $F$ is weekly nonlinear, i.e. it admits a representation of the form $F(q) =Uq+Phi(q)$, where $U$ is the isomorphism between the spaces $W^{ heta-1}_2 $ and $hat l_2^{ heta}$, and $Phi(q)$ is a compact map. Moreover we prove the estimate $ Phi(q) _{ au} leqslant C q _{ heta-1}$, where the value of $ au= au( heta)> heta$ is given explicitly and the constant $C$ depends only of the radius of the ball $ q _{ heta} leqslant R$ but does not depend on the function $q$, running through this ball
Source arXiv, math/0609213
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