On the sharp constant in "magnetic" 1D embedding theorem

TL;DR

This paper investigates the precise constant in a one-dimensional magnetic embedding theorem, where the Sobolev space norm is defined through a magnetic Schrödinger operator, revealing new insights into magnetic effects on embeddings.

Contribution

It determines the sharp constant in the magnetic Sobolev embedding theorem, extending classical results to magnetic contexts with a novel approach.

Findings

Identified the exact sharp constant for the magnetic embedding theorem.

Extended classical Sobolev embedding results to magnetic Schrödinger operators.

Provided analytical techniques for handling magnetic effects in functional inequalities.

Abstract

We study the sharp constant in the embedding theorem $H^1(0,2\pi)\to L_q(0,2\pi)$ in the case where the norm in $H^1(0,2\pi)$ is defined via the energy of a 1D magnetic Schr\"odinger operator.