Low energy resolvent estimates for slowly decaying attractive potentials

TL;DR

This paper establishes low energy resolvent estimates for Schrödinger operators with slowly decaying attractive potentials, utilizing an elementary commutator method to prove key spectral theorems without microlocal or functional-analytic tools.

Contribution

It introduces a simplified commutator approach to prove resolvent estimates and spectral theorems for slowly decaying attractive potentials in Schrödinger operators.

Findings

Proves Rellich's theorem for these potentials

Establishes the limiting absorption principle

Demonstrates Sommerfeld's uniqueness theorem

Abstract

We discuss the low energy resolvent estimates for the Schr\"odinger operator with slowly decaying attractive potential. The main results are Rellich's theorem, the limiting absorption principle and Sommerfeld's uniqueness theorem. For the proofs we employ an elementary commutator method due to Ito--Skibsted, for which neither of microlocal or functional-analytic techniques is required.