Abstract
We consider the linear Dirac operator with a (<svg style="vertical-align:-0.0pt;width:18.65px;" id="M1" height="10.6875" version="1.1" viewBox="0 0 18.65 10.6875" width="18.65" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6875)"> <g transform="translate(72,-63.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0">1</tspan> </text> </g> </g> </svg>)-homogeneous locally periodic potential that varies with respect to a small parameter. Using the notation of G-convergence for positive self-adjoint operators in Hilbert spaces we prove G-compactness in the strong resolvent sense for families of projections of Dirac operators. We also prove convergence of the corresponding point spectrum in the spectral gap.
Highlights
In the present work we study the asymptotic behavior of Dirac operators Hh with respect to a parameter h ∈ N as h → ∞
The paper is arranged as follows: in Section 2 we provide the reader with basic preliminaries on Dirac operators, G-convergence, and on the concepts needed from spectral theory
Given a Hilbert space X, let U, A be a measurable space for U ⊆ C and let A be a σ-algebra on U
Summary
In the present work we study the asymptotic behavior of Dirac operators Hh with respect to a parameter h ∈ N as h → ∞. We will study the asymptotic behavior the shifted perturbed Dirac operator Hh and the asymptotic behavior of the eigenvalues in the gap of the continuous spectrum of the shifted operator with respect to the perturbation parameter h. A detailed exposition of G-convergence for positive self-adjoint operators is found in Dal Maso 7. The Dirac operator is unbounded both from above and below This means that the theory of G-convergence for positive self-adjoint operators is not directly applicable toDirac operators. Journal of Function Spaces and Applications projections of Dirac operators which satisfy the positivity so that the theory of G-convergence becomes applicable.
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