We consider the Schrödinger operator \(H_0\) with constant magnetic field B of scalar intensity \(b>0\), self-adjoint in \(L^2({{\mathbb {R}}}^3)\), and its perturbations \(H_+\) (resp., \(H_-\)) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain \(\Omega _{\mathrm{in}} \subset {{\mathbb {R}}}^3\). We introduce the Krein spectral shift functions \(\xi (E;H_\pm ,H_0)\), \(E \ge 0\), for the operator pairs \((H_\pm ,H_0)\) and study their singularities at the Landau levels \(\Lambda _q : = b(2q+1)\), \(q \in {{\mathbb {Z}}}_+\), which play the role of thresholds in the spectrum of \(H_0\). We show that \(\xi (E;H_+,H_0)\) remains bounded as \(E \uparrow \Lambda _q\), \(q \in {{\mathbb {Z}}}_+\), being fixed, and obtain three asymptotic terms of \(\xi (E;H_-,H_0)\) as \(E \uparrow \Lambda _q\), and of \(\xi (E;H_\pm ,H_0)\) as \(E \downarrow \Lambda _q\). The first two divergent terms are independent of the perturbation, while the third one involves the logarithmic capacity of the projection of \(\Omega _{\mathrm{in}}\) onto the plane perpendicular to B.

中文翻译：

---

电磁哈密顿量的几何扰动的谱位移函数的阈值奇点

我们考虑具有标量强度\（b> 0 \）的恒定磁场B的Schrödinger算子\（H_0 \），在\（L ^ 2（{{\\ mathbb {R}}} ^ 3）\）中自伴，以及通过在有界域\（\ Omega _ {\ mathrm {in}）的边界上施加Dirichlet（resp。，Neumann）条件获得的扰动\（H _ + \）（resp。，\（H _- \））} \ subset {{\ mathbb {R}}} ^ 3 \）。我们为运算符对\（（H_ \ pm，H_0）\）引入了Kerin谱移函数\（\ xi（E; H_ \ pm，H_0）\），\（E \ ge 0 \）并研究它们在Landau层次上的奇点\（\ Lambda _q：= b（2q + 1）\），\（q \ in {{\ mathbb {Z}}} _ + \），在\（H_0 \）的频谱中起阈值的作用。我们表明，\（\ XI（E; H _ +，H_0）\）保持界定为\（E \ UPARROW \ LAMBDA _q \） ，\（Q \在{{\ mathbb {Z}}} _ + \） ，固定，并获得\（\ xi（E; H _-，H_0）\）的三个渐近项，如\（E \ uparrow \ Lambda _q \）和\（\ xi（E; H_ \ pm，H_0） \）作为\（E \ downarrow \ Lambda _q \）。前两个发散项与扰动无关，而第三个发散项涉及\（\ Omega _ {\ mathrm {in}} \）到垂直于平面的投影的对数容量乙。