03-110 Pavel Exner and Sylwia Kondej
Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in \$\mathbb{R}^3\$ (74K, LaTeX) Mar 13, 03
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Abstract. We investigate a class of generalized Schr\"{o}dinger operators in \$L^2(\mathbb{R}^3)\$ with a singular interaction supported by a smooth curve \$\Gamma\$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when \$\Gamma\$ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schr\"{o}dinger operator with a potential determined by the curvature of \$\Gamma\$. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if \$\Gamma\$ is not a straight line and the singular interaction is strong enough.

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