Bari-Markus property for Riesz projections of Hill operators with singular potentials

TL;DR

This paper investigates the spectral properties of Hill operators with singular potentials, demonstrating that the Riesz projections for large indices converge in Hilbert-Schmidt norm, indicating stability of the spectral decomposition.

Contribution

It establishes the Bari-Markus property for Riesz projections of Hill operators with $H^{-1}$ potentials, extending spectral stability results to singular potentials.

Findings

Sum of squared Hilbert-Schmidt norms converges for large n

Riesz projections approximate free operator projections

Spectral decomposition stability for singular potentials

Abstract

The Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,\pi],$ with $H^{-1}$ periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large $N,$ the Riesz projections $$ P_n = \frac{1}{2\pi i} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{z: |z-n^2|= n\} $$ are well defined. It is proved that $$\sum_{n>N} \|P_n - P_n^0\|^2_{HS} < \infty, $$ where $P_n^0$ are the Riesz projection of the free operator and $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm.