Derivatives of L^p eigenfunctions of Schrodinger operators

TL;DR

This paper establishes that for one-dimensional Schrödinger operators with certain potential conditions, eigenfunctions in L^p spaces have derivatives also in L^p, providing a new pointwise estimate for these derivatives.

Contribution

It introduces a novel pointwise L^p estimate for derivatives of eigenfunctions under specific potential assumptions, extending understanding of eigenfunction regularity.

Findings

Eigenfunctions in L^p imply their derivatives are also in L^p.

The results apply to potentials with negative parts in L^1_loc.

Provides a new regularity estimate for eigenfunctions of Schrödinger operators.

Abstract

Assuming the negative part of the potential is uniformly locally $L^1$, we prove a pointwise $L^p$ estimate on derivatives of eigenfunctions of one-dimensional Schrodinger operators. In particular, if an eigenfunction is in $L^p$, then so is its derivative, for $1\le p\le \infty$.