# Relative Spectral Invariants of Elliptic Operators on Manifolds

## TL;DR

This paper introduces new spectral invariants for pairs of elliptic operators on compact manifolds, capturing more geometric information than traditional invariants by depending on eigenvalues and eigensections.

## Contribution

It defines and analyzes novel relative spectral invariants for elliptic operators, including their short-time asymptotics and explicit coefficient calculations.

## Key findings

- Existence of short-time asymptotics for the new invariants
- Explicit formulas for the first two asymptotic coefficients
- Invariants depend on symbols of both operators

## Abstract

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prove the existence of the homogeneous short time asymptotics of the new invariants with the coefficients of the asymptotic expansion being integrals of some invariants that depend on the symbols of both operators. The first two coefficients of the asymptotic expansion are computed explicitly.