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Journal of the London Mathematical Society Advance Access published online on June 10, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp028
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© 2009 London Mathematical Society

On the energy of bound states for magnetic Schrödinger operators

Søren Fournais and Ayman Kachmar

Department of Mathematical Sciences
University of Aarhus
Ny Munkegade, Building 1530
DK-8000 Århus C
Denmark
fournais@imf.au.dk

We provide a leading order semiclassical asymptotics of the energy of bound states for magnetic Neumann Schrödinger operators in two-dimensional (exterior) domains with smooth boundaries. The asymptotics is valid all the way up to the bottom of the essential spectrum. When the spectral parameter is varied near the value where bound states become allowed in the interior of the domain, we show that the energy has a boundary and a bulk component. The estimates rely on coherent states, in particular on the construction of ‘boundary coherent states’, and magnetic Lieb–Thirring estimates.

2000 Mathematics Subject Classification 35P10, 35J10, 47F05, 81Q10.

The authors were supported by a Starting Independent Researcher grant by the ERC under the FP7. SF was also supported by the Danish Research Council and the Lundbeck Foundation.

Received October 22, 2008; revised March 11, 2009;
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