Quantum lattice transport along an infinitely extended perturbation

Abstract We consider a periodic quantum graph in the form of a rectangular lattice with the δ -coupling of strength γ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>≠</mml:mo> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> . We analyze the band spectrum of the system and show that it remains preserved as a set provided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> .