A Rellich type theorem for the generalized oscillator

TL;DR

This paper proves a Rellich type theorem for the generalized oscillator, characterizing the growth order of eigenfunctions using simple commutator methods without advanced analytical tools.

Contribution

Introduces a new proof of a Rellich type theorem for the generalized oscillator using elementary commutator techniques.

Findings

Eigenfunctions' growth order characterized

Proofs avoid energy cut-offs and microlocal analysis

Method simplifies analysis of the generalized oscillator

Abstract

For the generalized oscillator, we prove a Rellich type theorem, or characterize the order of growth of eigenfunctions. The proofs are given by an extensive use of commutator arguments invented recently by Ito and Skibsted. These arguments are simple and elementary and do not employ energy cut-offs or microlocal analysis.