On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems

Original · EN

In 1996, H. Volkmer observed that the inequality (∫₋₁¹1/|r||f'|dx)² ≤ K² ∫₋₁¹|f|²dx∫₋₁¹|(1/rf')'|²dx is satisfied with some positive constant K>0 for a certain class of functions f on [-1,1] if the eigenfunctions of the problem -y"=λ r(x)y, y(-1)=y(1)=0 form a Riesz basis of the Hilbert space L²|ᵣ|(-1,1). Here the weight r∈ L¹(-1,1) is assumed to satisfy xr(x)>0 a.e. on [-1,1]. We present two criteria in terms of Weyl-Titchmarsh m-functions for the Volkmer inequality to be valid. Using these results we show that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if r is odd.