Equiconvergence of spectral decompositions of Hill–Schrödinger operators

Djakov, Plamen Borissov and Mityagin, Boris (2013) Equiconvergence of spectral decompositions of Hill–Schrödinger operators. Journal of Differential Equations, 255 (10). pp. 3233-3283. ISSN 0022-0396

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Official URL: http://dx.doi.org/10.1016/j.jde.2013.07.030


We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = d(2)/dx(2) + v(x), x is an element of [0, pi], 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 parallel to S-N - S-N(0) : L-a -> L-b parallel to -> 0 if 1 < a <= b < infinity, 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 v is an element of H-alpha with 1/2 < alpha < 1 and 1/a = 3/2 - alpha, then we obtain uniform equiconvergence: parallel to S-N - S-N(0) : L-a -> L-infinity parallel to -> 0 as N -> infinity.

Item Type:Article
Uncontrolled Keywords:Hill-Schrodinger operators; Singular potentials; Spectral decompositions; Equiconvergence
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:22342
Deposited By:Plamen Borissov Djakov
Deposited On:06 Jan 2014 15:26
Last Modified:01 Aug 2019 12:04

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