An Lp Estimate for the Difference of Derivatives of Spectral Expansions Arising by One-dimensional Schroedinger Operators

Nebojša_L. Lažetić, Olivera_R. Đorđević

We prove the estimate $$ \|\sigma_{\mu}^{\prime}(x,f)- \tilde\sigma_{\mu}^{\prime}(x,f)\|_{L_p(G)}łe C\|f\|_{BV(G)}\cdot\mu^{1-1/p}, $$ where $2\le p<+\infty$, and $\sigma_{\mu}(x,f),\tilde \sigma_{\mu}(x,f)$ are the partial sums of spectral expansions of a function $f(x)\in BV(G)$, corresponding to arbitrary non-negative self-adjoint extensions of the operators $\Cal Lu=-u^{\prime\prime}+q(x)u$, $\tilde{\Cal L}u=-u^{\prime\prime}+\tilde q(x)u$ $(x\in G)$ respectively; the operators are defined on an arbitrary bounded interval $G\subset \Bbb R$.