Self‐adjointness of non‐semibounded covariant Schrödinger operators on Riemannian manifolds

Ognjen Milatovic

Mathematische Nachrichten, 2023, vol. 296, issue 9, 3967-3985

Abstract: In the context of a geodesically complete Riemannian manifold M, we study the self‐adjointness of ∇†∇+V$\nabla ^{\dagger }\nabla +V$, where ∇ is a metric covariant derivative (with formal adjoint ∇†$\nabla ^{\dagger }$) on a Hermitian vector bundle V$\mathcal {V}$ over M, and V is a locally square integrable section of EndV$\operatorname{End}\mathcal {V}$ such that the (fiberwise) norm of the “negative” part V−$V^{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number ε∈[0,1]$\varepsilon \in [0,1]$ and a positive function q on M satisfying certain growth conditions, such that ε∇†∇+V≥−q$\varepsilon \nabla ^{\dagger }\nabla +V\ge -q$, the inequality being understood in the quadratic form sense over Cc∞(V)$C_{c}^{\infty }(\mathcal {V})$. In the first result, which pertains to the case ε∈[0,1)$\varepsilon \in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case ε=1$\varepsilon =1$, we use the hyperbolic equation method.

Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202100252

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:296:y:2023:i:9:p:3967-3985

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().