Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectral gap of one-dimensional Schrödinger operators on large intervals

Joachim Kerner, Matthias Täufer

Abstract

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials which characterize its asymptotic behaviour.

On the spectral gap of one-dimensional Schrödinger operators on large intervals

Abstract

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials which characterize its asymptotic behaviour.

Paper Structure

This paper contains 3 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Model and results
  3. A shortcut in the proof of Theorem \ref{['ThmLowerBoundI']}

Key Result

Theorem 2.1

Let $v \in L^{\infty}(\mathbb{R})$ be a non-negative potential such that for some constant $C > 0$. Then for some constant $\beta$ and all $L$ large enough.

Theorems & Definitions (18)

  • Theorem 2.1: Upper bound I
  • proof
  • Remark 2.2
  • Theorem 2.3: Upper bound II
  • proof
  • Proposition 2.4: Unitary transformation
  • proof
  • Theorem 2.5: Lower bound
  • Remark 2.6
  • proof
  • ...and 8 more