关于存在由直线电流产生的磁场时拉普拉斯算子的基态
On the ground state of the Laplacian in presence of a magnetic field created by a rectilinear current
摘要：
我们考虑三维拉普拉斯算子与磁场所产生的无限大的直线电流轴承恒定电流。相关哈密顿量的谱是正半轴，因为带函数的无穷大范围都向0减小。我们使带函数在基态能量附近精确渐近，并表现出半经典行为。我们用电势扰动哈密顿量。通过对能带函数的分析，我们发现，对于慢衰减势，产生了无穷多个负本征值，而对于快衰减势，只出现了有限个本征值。我们的结果表明，在没有磁场的情况下，不同的边界类型的条件。
英文摘要：
We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band function near the ground energy and we exhibit a semi-classical behavior. We perturb the hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential, an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. Our results show different borderline type conditions that in the case where there is no magnetic field.
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