An $L_p$ estimate for the difference of derivatives of spectral expansions arising by one-dimensional Schrödinger operators

Abstract

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