# Non-harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

## TL;DR

This paper extends classical spectral and operator theory to non-harmonic boundary value problems on manifolds with boundary, providing criteria for operator compactness, boundedness, and spectral location, with applications to heat equations.

## Contribution

It introduces a non-harmonic version of Gohberg's Lemma, characterizes operator spectra via matrix analysis, and applies these results to evolution equations on manifolds with boundary.

## Key findings

- Provided explicit spectral information for pseudo-differential operators.
- Established criteria for operator compactness and boundedness in non-harmonic analysis.
- Applied results to ensure smoothness and stability of heat equation solutions.

## Abstract

In this paper, following the works on non-harmonic analysis of boundary value problems by Tokmagambetov, Ruzhansky and Delgado, we use Operator Ideals Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators, on a smooth manifold $\Omega$ with boundary $\partial \Omega$, in the context of the non-harmonic analysis of boundary value problems, introduced by Tokmagambetov and Ruzhansky in terms of a model operator $\mathfrak{L} $. Under certain assumptions about the eigenfunctions of the model operator, for symbols in the H\"ormander class $S^0_{1,0} (\overline{\Omega} \times \mathcal{I} )$, we provide a "non-harmonic version" of Gohberg's Lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a compact operator in $L^2(\Omega)$. Also, for pseudo-differential operators with symbols satisfying some integrability condition, one defines its associated matrix in terms of the biorthogonal system associated to $\mathfrak{L} $, and this matrix is used to give necessary and sufficient conditions for the $L^2(\Omega)$-boundedness, and to locate the spectrum of some operators. After that, we extend to the context of the non-harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations. Specifically we provide sufficient conditions to ensure the smoothness and stability of solutions to a generalised version of the heat equation.