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Asymptotic behavior of the resolvent of the Dirac operator. (English) Zbl 0879.47030

Demuth, Michael (ed.) et al., Mathematical results in quantum mechanics: International conference in Blossin (Germany), May 17-21, 1993. Basel: Birkhäuser Verlag. Oper. Theory, Adv. Appl. 70, 45-54 (1994).
Several results concerning the asymptotic behaviour of the resolvent of the Dirac operator are proved in this article. Dirac’s operator is defined by \(H=-i \sum^3_{j=1} \alpha_j {\partial \over\partial x_j} +\beta +Q(x)\), where \(\alpha_j\) and \(\beta\) are \(4\times 4\) Hermitian matrices satisfying the anticommutative relations and some additional assumptions and the \(4\times 4\) matrix \(Q\) satisfies a boundedness condition. The limiting absorption principle implies the existence of the extended resolvents \(R_\pm (\lambda)\) for \(\lambda\) real such that \(|\lambda |>1\). Some results concerning boundedness in norm and strong convergence to 0 for these and related operators are proved in the framework of weighted Sobolev spaces. Pseudodifferential operators also play an important role in the proofs.
For the entire collection see [Zbl 0791.00039].

MSC:

47F05 General theory of partial differential operators
47G30 Pseudodifferential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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