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.
Highlights
In recent years, non-selfadjoint operators are attracting increasing attention, both in view of applications to quantum mechanics and other branches of physics, and for the interesting mathematical challenges they present
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: σ (D0 + V ) = σ (D0) = R
The advantage of the last result lies in the explicit condition which is easy to check in applications
Summary
Non-selfadjoint operators are attracting increasing attention, both in view of applications to quantum mechanics and other branches of physics, and for the interesting mathematical challenges they present. The advantage of the last result lies in the explicit condition which is easy to check in applications In this result the eigenvalues are localized in an unbounded region around the continuous spectrum σ (D0) = (−∞, −m] ∪ [m, +∞) of the free Dirac operator D0. Remark 1.1 The crucial tool in our proof is a sharp uniform resolvent estimate for the free Dirac operator This approach is inspired by [22], where the result by Kenig et al [33] was used for the same purpose. This condition may not always be satisfied and depends on the norms of the potential V in the spaces L3(R3) and Y (R3) If this happens, the result in Theorem 1 and the one in [20] should be jointly taken in consideration for the eigenvalues bound.
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