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On Exact Estimates of Instability Zones of the Hill's Equation with Locally Integrable Potential

O. A. Veliev

TL;DR

This paper analyzes the one‑dimensional Hill (periodic Schrödinger) equation with a locally integrable real potential $q$ and derives exact high‑energy two‑term asymptotics for spectral gaps. Using Floquet theory and refined Fourier‑analytic expansions, the authors obtain sharp formulas for the eigenvalue shifts of the periodic and antiperiodic problems and express gap lengths $|\Delta_n|$ and $|\Omega_n|$ in terms of Fourier coefficients of $q$, including first and second terms (e.g., $|\Delta_n|\sim 2|q_{2n}|$ and $|\Delta_n|\sim 2|q_{2n}-S_{2n}+2Q_0Q_{2n}|$). They establish eigenfunction asymptotics and higher‑order expansions via $A_m$, $B_m$, and $E_{n,j,m}$, and illustrate the results with the Kronig–Penney model, where gap lengths exhibit distinct scaling ($|\Delta_k|\sim 1/k^2$ or $|\Omega_k|\sim 1/k$) depending on resonance. The work sharpens classical Titchmarsh estimates for $q\in L_1[0,1]$ and provides precise spectral data for applications in solid‑state models and instability analysis.

Abstract

In this paper we consider the one-dimensional Schrodinger operator L(q) with a periodic real and locally integrable potential q. We study the bands and gaps in the spectrum and explicitly write out the first and second terms of the asymptotic formulas for the length of the gaps in the spectrum.

On Exact Estimates of Instability Zones of the Hill's Equation with Locally Integrable Potential

TL;DR

This paper analyzes the one‑dimensional Hill (periodic Schrödinger) equation with a locally integrable real potential and derives exact high‑energy two‑term asymptotics for spectral gaps. Using Floquet theory and refined Fourier‑analytic expansions, the authors obtain sharp formulas for the eigenvalue shifts of the periodic and antiperiodic problems and express gap lengths and in terms of Fourier coefficients of , including first and second terms (e.g., and ). They establish eigenfunction asymptotics and higher‑order expansions via , , and , and illustrate the results with the Kronig–Penney model, where gap lengths exhibit distinct scaling ( or ) depending on resonance. The work sharpens classical Titchmarsh estimates for and provides precise spectral data for applications in solid‑state models and instability analysis.

Abstract

In this paper we consider the one-dimensional Schrodinger operator L(q) with a periodic real and locally integrable potential q. We study the bands and gaps in the spectrum and explicitly write out the first and second terms of the asymptotic formulas for the length of the gaps in the spectrum.
Paper Structure (3 sections, 15 theorems, 191 equations)

This paper contains 3 sections, 15 theorems, 191 equations.

Table of Contents

  1. Introduction and Preliminary Facts
  2. Main Results
  3. On the Kronig-Penney model

Key Result

Proposition 1

If $q\in L_{1}[0,1],$ then the following equalities hold and where $a_{1}((2\pi n)^{2}),$$a_{2}((2\pi n)^{2})$ and $b_{1}((2\pi n)^{2})$ are obtained respectively from $a_{1}(\lambda_{n,j}),$$a_{2}(\lambda_{n,j})$ and $b_{1}(\lambda_{n,j})$ by changing $\lambda_{n,j}$ to $(2\pi n)^{2}$ in the corresponding formulas.

Theorems & Definitions (18)

  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Theorem 5
  • Remark 1
  • ...and 8 more