We prove that the eigenvalues of the n-dimensional massive Dirac operator {mathscr {D}}_0 + V, nge 2, perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms L^1_{x_j} L^infty _{{widehat{x}}_j}, for jin {1,dots ,n}. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: sigma ({mathscr {D}}_0+V)=sigma ({mathscr {D}}_0)={mathbb {R}}. The main tools used are an abstract version of the Birman-Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

Eigenvalue bounds for non-selfadjoint Dirac operators / D’Ancona, Piero; Fanelli, Luca; Schiavone, Nico Michele. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - (2021). [10.1007/s00208-021-02158-x]

Eigenvalue bounds for non-selfadjoint Dirac operators

D’Ancona, Piero
;
Fanelli, Luca;Schiavone, Nico Michele
2021

Abstract

We prove that the eigenvalues of the n-dimensional massive Dirac operator {mathscr {D}}_0 + V, nge 2, perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms L^1_{x_j} L^infty _{{widehat{x}}_j}, for jin {1,dots ,n}. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: sigma ({mathscr {D}}_0+V)=sigma ({mathscr {D}}_0)={mathbb {R}}. The main tools used are an abstract version of the Birman-Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.
2021
Dirac operator; non-selfadjoint perturbation; localization of eigenvalues; Birman-Schwinger principle
01 Pubblicazione su rivista::01a Articolo in rivista
Eigenvalue bounds for non-selfadjoint Dirac operators / D’Ancona, Piero; Fanelli, Luca; Schiavone, Nico Michele. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - (2021). [10.1007/s00208-021-02158-x]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1492616
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