Equiconvergence of spectral decompositions of Hill-Schrödinger operators

Abstract

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L=-d^2/d{x}^2+v(x),$ $x\in [0,\pi ],$ with $H_{per}^{-1}$-potential and the free operator $L^0=-d^2/d{x}^2,$ subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $$\Vert S_N-S_N^0:L^a\to L^b\Vert \to 0\text{if}1<a\le b<\infty ,1/a-1/b<1/2,$$ where $S_N$ and $S_N^0$ are the $N$-th partial sums of the spectral decompositions of $L$ and $L^0.$ Moreover, if $v\in H^{-\alpha }$ with $1/2<\alpha <1$ and $\frac{1}{a}=(3/2)-\alpha ,$ then we obtain uniform equiconvergence: $\Vert S_N-S_N^0:L^a\to L^{\infty }\Vert \to 0$ as $N\to \infty .$

References in corpus (5)

• Inverse spectral problems for Sturm-Liouville operators with singular potentials
• Schrödinger operators with singular Gordon potentials
• Riesz bases consisting of root functions of 1D Dirac operators
• Equiconvergence theorems for differential operators
• Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions