Equiconvergence of spectral decompositions of Hill-Schr\"odinger operators

TL;DR

This paper investigates the conditions under which the spectral decompositions of Hill operators with singular potentials converge uniformly to those of the free operator, expanding understanding of spectral behavior in various functional spaces.

Contribution

It establishes new convergence results for spectral decompositions of Hill operators with $H_{per}^{-1}$ potentials, including uniform convergence in certain Sobolev spaces.

Findings

Spectral decompositions converge in operator norm under specified conditions.

Uniform equiconvergence is achieved for potentials in $H^{-eta}$ with $1/2<eta<1$.

Convergence criteria depend on the relationship between $a$ and $b$ in $L^a$ to $L^b$ mappings.

Abstract

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x), $ $x \in [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 $$ \|S_N - S_N^0: L^a \to L^b \| \to 0 \quad \text{if} \;\; 1<a \leq b< \infty, \;\; 1/a - 1/b <1/2, $$ 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: $\|S_N - S_N^0: L^a \to L^\infty \| \to 0 $ as $N \to \infty. $