谱奇异性、双正交正规系统和双参数复点相互作用族
Spectral Singularities, Biorthonormal Systems, and a Two-Parameter Family of Complex Point Interactions
摘要：
复散射势v（x）的一个奇怪的特征是谱奇异性的出现。我们提供了一个定量的描述谱奇异性，确定他们的障碍，存在一个完整的双正交系统组成的本征函数的哈密顿算子，即，- \frac{d^2}{dx^2} +v（x）及其伴随。我们建立了这种描述与数学家关于势函数v（x）=z_-δ（x+a）+z_+δ（x-a）的谱奇点定义的等价性，其中z_ \pm和a分别是复参数和实参数，δ（x）是Dirac δ函数.我们提供了一个通过分析这种潜在的光谱特性，并确定在空间中的耦合常数z_ \pm的区域，它允许的束缚态和光谱奇异性。特别是，我们找到了一个明确的限制在某些地区的哈密顿量是准厄米特的大小，并检查施加PT对称性的后果。
英文摘要：
A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., -\frac{d^2}{dx^2}+v(x), and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v(x)=z_-δ(x+a)+z_+δ(x-a), where z_\pm and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z_\pm where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
