Bound states of a converging quantum waveguide

Cardone, Giuseppe; Nazarov, Sergei A.; Ruotsalainen, Keijo

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 305-315.

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 - ε, where ε > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weakly coupled bound state below π², the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when ε → 0⁺.

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