Eigenvalue bounds for non-selfadjoint Dirac operators

Abstract

We prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 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 $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , 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: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = 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.

Keller-type bounds for Dirac operators perturbed by rigid potentials

“…Instead, we are mainly devoted to the research of a compact region in which to localize the point spectrum. Some progress in this direction are achieved by the second author together with D'Ancona and Fanelli in [18] and D'Ancona, Fanelli and Krejčiřík in [17]. In the first work, we proved a result that generalizes in higher dimensions the previous one by Cuenin, Laptev and Tretter [12].…”

“…The above theorem is a generalization of Theorem 7, dropping the many restrictions on V and with slightly modified definitions of centers and radius of the disks. In some sense, it can be seen as the radial version of the result in [18] recalled above in (1.8)-(1.9).…”

“…The combination of the Birman-Schwinger principle with resolvent estimates for free operators is one of the way to approach the localization problem for eigenvalues: it has been widely employed in the later times (see e.g. [26,12,19,30,10,25,23,4,5,18,17] among others) and it will be the approach we are going to follow in this work too, as we will see. However, despite the robustness of the Birman-Schwinger principle, it is not the only tool one could use to obtain spectral enclosures for non-selfadjoint operators: another powerful technique is the method of multipliers, see e.g.…”

Localization of eigenvalues for non-self-adjoint Dirac and Klein–Gordon operators

“…To make up for this lack of conventional methods, the key tool usually employed in the non-selfadjoint settings is the celebrated Birman-Schwinger principle (see, e.g., [1][2][3][4][5][6][7][8]8,9] to cite just few recent works).…”

Localization of eigenvalues for non-self-adjoint Dirac and Klein-Gordon operators

“…To make up for this lack of conventional methods, the key tool usually employed in the non-selfadjoint settings is the celebrated Birman-Schwinger principle (see, e.g., [20,10,16,21,9,18,17,3,14,3] to cite just few recent works). Roughly speaking (see below for precise statements), the principle states that z is an eigenvalue of an operator H := H 0 + B * A if and only if −1 is an eigenvalue of the Birman-Schwinger operator K z := A(H 0 − z) −1 B * .…”