Strong magnetic fields,Dirichlet boundaries,and spectral gaps |
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Authors: | Rainer Hempel Ira Herbst |
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Institution: | 1. Erwin Schr?dinger International Institute for Mathematical Physics, Pasteurgasse 4/7, A-1090, Vienna, Austria
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Abstract: | We consider magnetic Schrödinger operators $$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$ inL 2(R n ), where $\vec a \in C^1 (R^n ;R^n )$ and λεR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with $\vec a$ , and $M_{\vec a} = \{ x;\vec a(x) = 0\}$ , we prove that $H(\lambda \vec a)$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set $M\backslash M_{\vec a}$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $M\backslash M_{\vec a}$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of $H(\lambda \vec a)$ . We finally address the question of absolute continuity of $\vec a$ for periodic $H(\vec a)$ . |
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