Spectral Geometry and the Semiclassical Approach

TL;DR

This paper explores the deep analogy between Spectral Geometry and Semiclassical Theory, providing new insights, practical formulas, and examples related to quantum field theory, spectral variations, and inverse problems.

Contribution

It introduces a unified perspective linking Spectral Geometry and Semiclassics, with new formulas and methods for spectral variations and inverse problems.

Findings

Derived practical formulas for spectral variations in Robin problems

Established connections between spectral geometry and semiclassical analysis

Provided examples related to quantum field theories like QCD and String Theory

Abstract

The paper is dedicated to the close analogy between these two theories - some problems lying at the very root of Spectral Geometry are viewed in the context of Semiclassics, and vise versa. The treatment starts from a very basic level and gradually goes into the deep, ending up with some rather up to date results. There are many examples concerning effective actions in QCD and String Theory, functional determinants, first integrals from asymptotic expansions, path quantization etc. The major theorems are only given brief explanations and simple constructive proofs, but some rather practical formulae are obtained in return. There is a small exercise at the end - obtaining explicit expressions for the spectral variations of the Robin problem for a planar domain. The final result is purely in the spirit of the familiar Sationary Perturbation Theory. The idea for investigating this particular problem (in the context of an old exact inverse result obtained by Guillemin and Melrose), very much motivating most of the work, belongs to my scientific advisor Dr. Petko Nikolov.