Gap asymptotics in a weakly bent leaky quantum wire (original) (raw)
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Reviews in Mathematical Physics, 2004
We investigate a class of generalized Schrödinger operators in L2(ℝ3) with a singular interaction supported by a smooth curve Γ. We find a strong-coupling asymptotic expansion of the discrete spectrum in the case when Γ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of Γ. In the same way, we obtain asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if Γ is not a straight line and the singular interaction is strong enough.