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On perturbations of the spectrum of one-dimensional PT-symmetric periodic Schrodinger operator

P. G. Grinevich, I. A. Taimanov

TL;DR

This work analyzes how small PT-symmetric perturbations of a periodic one-dimensional Schrödinger operator affect the Bloch spectrum and the divisor of zeros of the Bloch function. Using first-order perturbation theory within the algebro-geometric finite-gap framework, it yields explicit leading-order expressions for the perturbed spectrum near resonances and for the dynamics of the Bloch divisor. The main findings show that resonant points split into spectral gaps when $c_n c_{-n} > 0$, or form complex-branch spectrum with endpoints $E_n^{\pm}$ when $c_n c_{-n} < 0$, with the divisor projections becoming ellipses whose foci are the spectral-curve branch points. These results provide a concrete leading-order description of lacuna analogs in PT-symmetric periodic systems and connect spectral deformation to algebro-geometric data relevant to inverse spectral analysis.

Abstract

For PT-symmetric periodic Schrodinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of the perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.

On perturbations of the spectrum of one-dimensional PT-symmetric periodic Schrodinger operator

TL;DR

This work analyzes how small PT-symmetric perturbations of a periodic one-dimensional Schrödinger operator affect the Bloch spectrum and the divisor of zeros of the Bloch function. Using first-order perturbation theory within the algebro-geometric finite-gap framework, it yields explicit leading-order expressions for the perturbed spectrum near resonances and for the dynamics of the Bloch divisor. The main findings show that resonant points split into spectral gaps when , or form complex-branch spectrum with endpoints when , with the divisor projections becoming ellipses whose foci are the spectral-curve branch points. These results provide a concrete leading-order description of lacuna analogs in PT-symmetric periodic systems and connect spectral deformation to algebro-geometric data relevant to inverse spectral analysis.

Abstract

For PT-symmetric periodic Schrodinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of the perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.
Paper Structure (3 sections, 2 theorems, 34 equations, 1 figure)

This paper contains 3 sections, 2 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary facts
  3. Main results

Key Result

Theorem 1

For the small ${\mathcal{PT}}$-perturbations eq:sch3 of the zero potential the resonant point eq:sch6 of the spectrum of $L$ 1) is opened into a gap for $c_n c_{-n} >0$; 2) is opened into a branch of the spectrum which is a curve transversal to the spectral half-line $\{E >0\}$ and bounded by the co

Figures (1)

  • Figure 1: At the top: the case $c_nc_{-n}>0$. The perturbation generates a gap in the spectrum. In the middle: the case $c_nc_{-n}=0$. In the leading order of the perturbation theory the spectral curve has a double point. At the bottom: the case $c_nc_{-n}<0$. The is no gap in the spectrum, moreover, the spectrum contains an interval perpendicular to the real line.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2