Strong magnetic fields,Dirichlet boundaries,and spectral gaps

We consider magnetic Schrödinger operators $$H(lambda vec a) = ( - inabla - lambda vec a(x))^2$$ 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 $Mbackslash M_{vec a}$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $Mbackslash M_{vec a}$ 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 $H(vec a)$ .