势偏置下的漏面谱
On the spectrum of leaky surfaces with a potential bias
摘要：
我们讨论了$H = -Δ+ V(x) - αδ(x-Σ)$型算子的吸引相互作用，$α>0$，在$L^2(\mathbb{R}^3)$，其中$Σ$是一个无限的表面，渐近平面和光滑的紧致外，划分空间为两个区域，其中一个是凸的，和$V$是一个潜在的偏见是一个积极的常数$V_0$在一个区域和零的其他。我们找到了基本的谱，并询问了离散谱的存在性，特别注意临界情况，$V_0=α^2$。我们表明，$σ_\mathrm{disc}(H)$是空的，如果偏见是支持在“外部”区域，而在相反的情况下，孤立的特征值可能存在。
英文摘要：
We discuss operators of the type $H = -Δ+ V(x) - αδ(x-Σ)$ with an attractive interaction, $α>0$, in $L^2(\mathbb{R}^3)$, where $Σ$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $V$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0=α^2$. We show that $σ_\mathrm{disc}(H)$ is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist.
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