, Volume 86, Issue 1, pp 542-544

Equiconvergence of spectral decompositions of Hill operators

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Abstract

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = −d 2/dx 2 + v(x), xL 1([0, π], with H per −1 -potential and the free operator L 0 = −d 2/dx 2, subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$ , where S N and S N 0 are the N-th partial sums of the spectral decompositions of L and L 0. Moreover, if vH −α with 1/2 < α < 1 and $\frac{1} {a} = \frac{3} {2} - \alpha $ , then we obtain the uniform equiconvergence ‖S N S N 0 : L a L ‖ → 0 as N → ∞.

Original Russian Text © P.B. Djakov, B.S. Mityagin, 2012, published in Doklady Akademii Nauk, 2012, Vol. 445, No. 5, pp. 498–500.
Presented by Academician S. P. Novikov January 15, 2012 Received
The article was translated by the authors.