The spectrum of magnetic Schr\"odinger operators and $k$-form Laplacians on conformally cusp manifolds
Golénia, Sylvain ; Moroianu, Sergiu
arXiv, 0507443 / Harvested from arXiv
Text on arXiv
We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an integral relative cohomology class. It follows that the essential spectrum is not stable by perturbation even by a compactly supported magnetic field. We also treat magnetic operators perturbed with electric fields. In the same context we describe the essential spectrum of the $k$-form Laplacian. This is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. In all the cases when we have pure-point spectrum we give Weyl-type asymptotics for the eigenvalue-counting function. In the other cases we describe the essential spectrum.
Publié le : 2005-07-21
Classification:  Mathematics - Differential Geometry,  Mathematical Physics,  Mathematics - Spectral Theory,  58J40,  58Z05 
@article{0507443,
     author = {Gol\'enia, Sylvain and Moroianu, Sergiu},
     title = {The spectrum of magnetic Schr\"odinger operators and $k$-form Laplacians
  on conformally cusp manifolds},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0507443}
}

Golénia, Sylvain; Moroianu, Sergiu. The spectrum of magnetic Schr\"odinger operators and $k$-form Laplacians
  on conformally cusp manifolds. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0507443/