# Schatten class conditions for functions of Schr\"odinger operators

## TL;DR

This paper establishes sharp conditions under which the difference of functions of free and perturbed Schrödinger operators belongs to Schatten classes, depending on potential decay and function smoothness, including unbounded functions for certain p.

## Contribution

It provides a precise criterion linking potential decay, function smoothness, and Schatten class membership for Schrödinger operator functions.

## Key findings

- Derived sharp Schatten class conditions based on potential decay and function smoothness.
- Extended results to include some unbounded functions for p > 1.
- Connected Schatten class membership to Besov space regularity.

## Abstract

We consider the difference $f(H_1)-f(H_0)$, where $H_0=-\Delta$ and $H_1=-\Delta+V$ are the free and the perturbed Schr\"odinger operators in $L^2(\mathbb R^d)$, and $V$ is a real-valued short range potential. We give a sharp sufficient condition for this difference to belong to a given Schatten class $\mathbf S_p$, depending on the rate of decay of the potential and on the smoothness of $f$ (stated in terms of the membership in a Besov class). In particular, for $p>1$ we allow for some unbounded functions $f$.