Renormalization and Essential Singularity

TL;DR

This paper explores how the behavior of functions under differentiation impacts renormalization theory, especially in quantum gravity, by classifying potential singularities based on eigenfunction properties.

Contribution

It introduces a classification of singularities in potentials for Schrödinger equations considering eigenfunction analyticity, highlighting implications for renormalization.

Findings

Singularity classification depends on eigenfunction analyticity.

Differentiation behavior of functions affects renormalization assumptions.

Potential singularities are characterized by eigenfunction properties near zeros.

Abstract

In usual dimensional counting, momentum has dimension one. But a function f(x), when differentiated n times, does not always behave like one with its power smaller by n. This inevitable uncertainty may be essential in general theory of renormalization, including quantum gravity. As an example, we classify possible singularities of a potential for the Schr\"{o}dinger equation, assuming that the potential V has at least one $C^2$ class eigen function. The result crucially depends on the analytic property of the eigen function near its 0 point.