Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in $\mathbb{R}^3$
Exner, P. ; Kondej, S.
arXiv, 0303033 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2003-03-13
Classification:  Mathematical Physics,  Condensed Matter,  Quantum Physics 
@article{0303033,
     author = {Exner, P. and Kondej, S.},
     title = {Strong-coupling asymptotic expansion for Schr\"odinger operators with a
  singular interaction supported by a curve in $\mathbb{R}^3$},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0303033}
}

Exner, P.; Kondej, S. Strong-coupling asymptotic expansion for Schr\"odinger operators with a
  singular interaction supported by a curve in $\mathbb{R}^3$. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0303033/