# Spectral asymptotics for infinite order pseudo-differential operators

## TL;DR

This paper investigates the spectral properties of infinite order pseudo-differential operators, revealing that their spectral asymptotics differ from finite order cases by exhibiting log-type behavior, and addresses the complexities of their ultradistributional setting.

## Contribution

It provides the first detailed analysis of spectral asymptotics for infinite order pseudo-differential operators in an ultradistributional framework.

## Key findings

- Spectral counting functions exhibit log-type asymptotics.
- Finite order Weyl calculus is inadequate for infinite order operators.
- The study advances understanding of spectral behavior in complex operator classes.

## Abstract

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.