We consider magnetic Schrödinger operator H = (i ∇ + A)² - αδ_Γ with an attractive singular interaction supported by a piecewise smooth curve Γ being a local deformation of a straight line. Such a Hamiltonian describes a charged particle confined to a leaky quantum wire and exposed to the magnetic field B which is supposed to be nonzero and local. We show that the essential spectrum of this system is $\left[-\frac{1}{4}{\alpha }^2,\infty \right]$, the same as for the nonmagnetic operator with a straight Γ, and demonstrate a sufficient condition for the discrete spectrum of H to be empty.

中文翻译：

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磁场对局部变形漏线离散谱的影响

我们考虑磁性薛定谔算子H = ( i ∇ + A ) ² - αδ _Γ具有吸引力的奇异相互作用，由分段平滑曲线Γ支持，该曲线是直线的局部变形。这样的哈密顿量描述了一个带电粒子被限制在泄漏的量子线中并暴露在磁场B 中，该磁场B应该是非零的和局部的。我们证明了这个系统的本质谱是 $\left[——\frac{1}{4}{\alpha }^2,\infty \right]$，与具有直线 Γ 的非磁性算子相同，并证明了H的离散谱为空的充分条件。