Bound states in a locally deformed waveguide: the critical case
Exner, P. ; Vugalter, S. A.
arXiv, 9601002 / Harvested from arXiv
Text on arXiv
We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one straight boundary and a width $\,a(1+\lambda f(x))\,$, where $\,f\,$ is a smooth function of a compact support with a length $\,2b\,$. We show that in the critical case, $\,\int_{-b}^b f(x)\, dx=0\,$, the operator has no bound states for small $\,|\lambda|\,$ if $\,b<(\sqrt{3}/4)a\,$. On the other hand, a weakly bound state exists provided $\,\|f'\|< 1.56 a^{-1}\|f\|\,$; in that case there are positive $\,c_1, c_2\,$ such that the corresponding eigenvalue satisfies $\,-c_1\lambda^4\le \epsilon(\lambda)- (\pi/a)^2 \le -c_2\lambda^4\,$ for all $\,|\lambda|\,$ sufficiently small.
Publié le : 1996-01-23
Classification:  Mathematics - Functional Analysis,  Condensed Matter,  Mathematical Physics,  Quantum Physics 
@article{9601002,
     author = {Exner, P. and Vugalter, S. A.},
     title = {Bound states in a locally deformed waveguide: the critical case},
     journal = {arXiv},
     volume = {1996},
     number = {0},
     year = {1996},
     language = {en},
     url = {http://dml.mathdoc.fr/item/9601002}
}

Exner, P.; Vugalter, S. A. Bound states in a locally deformed waveguide: the critical case. arXiv, Tome 1996 (1996) no. 0, . http://gdmltest.u-ga.fr/item/9601002/