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

Abstract

We prove the estimate $$\Vert {\sigma }_{\mu }^{\prime }(x,f)-{\tilde{\sigma }}_{\mu }^{\prime }(x,f){\Vert }_{L_p(G)}\le C\Vert f{\Vert }_{BV(G)}\cdot {\mu }^{1-1/p},$$ 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 $f(x)ınBV(G)$, corresponding to arbitrary non-negative self-adjoint extensions of the operators $\mathcal{L}\mathcal{u}\mathcal{=}\mathcal{-}{\mathcal{u}}^{\prime \prime }\mathcal{+}\mathcal{q}\mathcal{(}\mathcal{x}\mathcal{)}\mathcal{u}$, $\tilde{\mathcal{L}}u=-u^{\prime \prime }+\tilde{q}(x)u$ $(xınG)$ respectively; the operators are defined on an arbitrary bounded interval $G\subset \mathbb{R}$.