Three-dimensional magnetic Schr\"odinger operator with the potential supported in a tube

In this paper, we study the following magnetic Schr\"odinger operator in $\mathbb{R}^3$: $$H=(i \nabla +A)^2- \tilde{V},$$ where $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and $B:=\mathrm{rot}(A)$ is a non-zero and local (i.e., a compact supported) magnetic field. Based on some new strategies, we first prove that the magnetic field does not change the essential spectrum of this system. Finally, in the last section of this paper, we establish the sufficient condition such that the discrete spectrum is empty.