Bari-Markus property for Riesz projections of Hill operators with 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.