Eigenvalue bounds for non-selfadjoint Dirac operators
Piero D’Ancona, Luca Fanelli, Nico Michele Schiavone
Abstract
Abstract We prove that the eigenvalues of the n -dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> , $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mn>1</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:msub> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , for $$j\in \{1,\dots ,n\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . 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: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . 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.