具有复稀疏势的薛定谔算子
Schrödinger operators with complex sparse potentials
摘要：
我们建立了薛定谔算子与复位势，满足某种弱形式的稀疏性定量的上限和下限。我们的第一个结果是一个例子的定量版本，由于S。Boegli（Comm. Math. Phys.，2017，352，629-639），其特征值累积到本质谱的每个点。第二个结果表明，Frank（Bull. Lond.数学会，2011，43，745-750和Trans. Amer. Math. Soc.，2018年，370，219-240）可以改善稀疏的潜力。第三个结果推广了Klaus（Ann.Inst.H. Poincaré Sect. A（N.S.），1983，38，7-13）关于多维非自伴情况的本质谱的表征。第四个结果表明，在一维中，纯虚（非稀疏）阶跃势具有出乎意料的许多本征值，与共振的数量相当。我们的例子表明，几个已知的上界是尖锐的。
英文摘要：
We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys., 2017, 352, 629-639), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull. Lond. Math. Soc., 2011, 43, 745-750 and Trans. Amer. Math. Soc., 2018, 370, 219-240) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann. Inst. H. Poincaré Sect. A (N.S.), 1983, 38, 7-13) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.
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