Fermi’s golden rule in tunneling models with quantum waveguides perturbed by Kato class measures

Abstract In this paper we consider two dimensional quantum system with an infinite waveguide of the width d and a transversally invariant profile. Furthermore, we assume that at a distant ρ there is a perturbation defined by the Kato measure. We show that, under certain conditions, the resolvent of the Hamiltonian has the second sheet pole which reproduces the resonance at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with the asymptotics <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mrow> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">|</mml:mo> </mml:msqrt> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>ρ</mml:mi> </mml:mfrac> <mml:mrow> <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for ρ large and with the resonant energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . Moreover, we show that the imaginary component of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> satisfies Fermi’s golden rule which we explicitly derive.