Uniconvergence theorems for Sturm--Liouville operators with potentials from Sobolev space W₂^-1[0,π]

2008/06/18 by Sadovnichaya, I. V.
#34L10 #34L40 #47E05 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0806.3016

Abstract

We consider a Sturm--Liouville Ly=-y''+q(x)y in space L₂[0,π] with potential from Sobolev space W₂^-1[0,π]. Moreover, we assume, that q=u', where u∈ L₂[0,π]. We consider Direchlet boundary conditions y(0)=y(π)=0, although we can treat a boundary conditions of Sturm type. It is known, that operators of such class have a discrete spectr with only accumulation point +∞ and the system \y_k\₁^∞ of eigen and associated functions is a Riesz basis in L₂[0,π]. Moreover, this basis is a Hilbert--Schmidt perturbation of the basis \sin(kx)\₁^∞. In this paper we prove the uniconvergence theorem: for any element f∈ L₂[0,π] the sequence P_nf-S_nf→0 as n→∞ in C[0,π] (here P_n and S_n are the Riesz projectors to \y_k\₁ⁿ and \sin(kt)\₁ⁿ respectively).

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