Weyl-function asymptotics with applications to $$\mathcal{P}\mathcal{T}$$-symmetry and point interactions

Abstract Weyl m -function characterisations can be used to describe the behaviour of eigenvalues of Schrödinger operators with point interactions. Using complex scaling, transformators and a local Borg-Marchenko theorem we describe the asymptotic behaviour of $$m(\lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> as $$|\lambda |\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> in different regions of $$\mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> . Our asymptotic results are applied to the evolution of real eigenvalues of $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> and $$\delta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> perturbations of $$\mathcal{P}\mathcal{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> -symmetric problems as functions of coupling constants.