A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schr\"odinger Operators with Periodic and Antiperiodic Boundary Conditions

TL;DR

This paper establishes criteria for when non-self-adjoint Schrödinger operators with periodic boundary conditions have Riesz or Schauder bases of root vectors, extending spectral basis theory to these operators.

Contribution

It provides necessary and sufficient conditions for Riesz basis property and discusses Schauder basis criteria for such operators with potentials in L^2 and L^p spaces.

Findings

Derived criteria for Riesz basis of root vectors in non-self-adjoint Schrödinger operators.

Extended basis theory to operators with potentials in L^2 and L^p spaces.

Analyzed the case of Schauder basis for operators in L^p spaces.

Abstract

Under the assumption that $V \in L^2([0,\pi]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schr\"odinger operators $-d^2/dx^2+V$ in $L^2([0,\pi]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schr\"odinger operators $-d^2/dx^2+V$ in $L^p([0,\pi]; dx)$, $p \in (1,\infty)$.