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How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that $$ \left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||B||_{L^\infty(0,L)} $$ using the Courant-Fischer min-max principle?

Also, is it true that the corresponding eigenfunction form a hortonormal basis of $L^2$?

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    $\begingroup$ Does this answer your question? Asymptotic behavior of Sturm-Liouville eigenvalues $\endgroup$
    – mcd
    Commented Dec 14, 2020 at 14:30
  • $\begingroup$ @mcd Thanks! This helps a lot, but it does not explain the key point: how do you prove $$\lambda_n(B)+\lambda_{\min}(C)\le\lambda_n(B+C)\le\lambda_n(B)+\lambda_{\max}(C) \ ?$$ $\endgroup$
    – Lao
    Commented Dec 14, 2020 at 14:39
  • $\begingroup$ @mcd Also, is it true that the corresponding eigenfunction form a basis of $L^2$? $\endgroup$
    – Lao
    Commented Dec 15, 2020 at 17:30

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