Eigenvalue Asymptotics Near a Flat Band in the Presence of a Slowly Decaying Potential

We provide eigenvalue asymptotics for a Dirac-type operator on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mspace width=".1em"/> <mml:mn>2</mml:mn> </mml:math> , perturbed by multiplication operators that decay as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>μ</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msup> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>γ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> </mml:math> . We show that the eigenvalues accumulate near the value of the flat band at a “semiclassical” rate with a constant that encodes the structure of the flat band. Similarly, we show that this behavior can be obtained also for a Laplace operator on a periodic graph.