A note on semistable unitary operators on $L^2(\mathbb{R})$

TL;DR

This work characterizes semistable unitary operators on $L^2(\mathbb{R})$ under translation-invariance, symmetry, and local uniform continuity (LUC) under dilation, showing they act as Fourier multipliers with phase $e^{i\beta|\xi|^\alpha}$ and thus generate Schrödinger evolutions of fractional order $\alpha$. It further analyzes weakly measurable operator groups consisting of such operators, proving each time-parameter member has the same spectral form and giving a group-law representation $\widehat{T(t)f}(\xi)=e^{i\beta t|\xi|^\alpha}\widehat{f}(\xi)$, identifying the Schrödinger group of order $\alpha$. The results unify semistability, dilation-geometry, and unitary group structure to classify fractional-order Schrödinger-type dynamics on the real line. The findings provide a clear operator-theoretic description of fractional quantum evolutions on $L^2(\mathbb{R})$ and establish uniqueness of the defining parameters when the operator is nontrivial. Overall, the note connects semistability with explicit spectral multipliers and their associated one-parameter groups, offering a precise framework for fractional-order unitary dynamics.

Abstract

In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under dilation. As a consequence, we characterize one-parameter groups formed by such operators, which are of the form $e^{iβt|{d}/{dx}|^α}$, with $α,β\in\mathbb R$.

Theorems & Definitions (30)

• Definition 1.1: semistable operator
• Definition 1.2: local uniform continuity, LUC
• Definition 1.3
• Definition 1.4
• Theorem 1.1
• Definition 1.5: Schrödinger operator of order $\alpha$
• Corollary 1.1
• Theorem 1.2
• Definition 1.6: Schrödinger group of order $\alpha$
• Corollary 1.2