Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

Boldt, Sebastian; Güneysu, Batu

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 3, pp. 785-829

Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators ${𝒲}_{\pm }(D_h,D_g,I_{g,h})$ and ${𝒲}_{\pm }(D_h^2,D_g^2,I_{g,h})$, where $I_{g,h}$ is the canonically given unitary map between the underlying $L^2$-spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.

Étant donné une variété spin non-compacte $M$ avec une structure spinorielle topologique fixée et deux métriques riemanniennes complètes $g$ et $h$ sur $M$ à courbures sectionnelles bornées, nous prouvons un critère d’existence et de complétude des opérateurs d’onde ${𝒲}_{\pm }(D_h,D_g,I_{g,h})$ et ${𝒲}_{\pm }(D_h^2,D_g^2,I_{g,h})$, où $I_{g,h}$ est l’application unitaire canoniquement donnée entre les espaces $L^2$ de spineurs sous-jacents. Ce critère ne requiert aucune hypothèse de rayon d’injectivité et amène à un critère de stabilité du spectre absolument continu d’un opérateur de Dirac et de son carré sous un flot de Ricci.

Classification : 35P25, 53C27, 58J65

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