具有超表面磁阱的周期薛定谔算子的谱隙：底部附近的分析
Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom
摘要：
考虑非紧黎曼流形$M$上的周期磁薛定谔算子$H^h$，它依赖于半经典参数$h>0$，使得$H^1(M, {\mathbb R})=0$具有离散群的适当不连续的余紧等距作用量.我们假设没有电场，磁场中有一组周期性的致密磁阱。我们假设磁场在超曲面$S$上规则地消失。首先，我们证明了$L^2(M)$中算子$H^h$的谱的底部$λ_0(H^h)$的上下估计。然后，假设$S$上的约化谱问题存在非简并小阱，证明了在接近$λ_0(H^h)$的区域中，算子$H^h$存在任意大数目的谱隙，如$h\to 0$ .在这种情况下，我们也得到了一个井问题的特征值的上估计。
英文摘要：
We consider a periodic magnetic Schrödinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a noncompact Riemannian manifold $M$ such that $H^1(M, {\mathbb R})=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface $S$. First, we prove upper and lower estimates for the bottom $λ_0(H^h)$ of the spectrum of the operator $H^h$in $L^2(M)$. Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on $S$, we prove the existence of an arbitrary large number of spectral gaps for the operator $H^h$ in the region close to $λ_0(H^h)$, as $h\to 0$. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.
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