奇异一维束缚态问题及其退化
A Singular One-Dimensional Bound State Problem and its Degeneracies
摘要：
本文将一维$N$吸引狄拉克δ势系统的束缚态问题表述为一个$N \times N$矩阵本征值问题（$ΦA =ωA$）.本文的主要目的是说明一维非简并性定理在等距分布的Dirac δ势中失效，其中矩阵$Φ$成为循环矩阵的一种特殊形式.然后利用Perron-Frobenius定理给出了基态总是非简并的和相应的波函数可以取正值的初等证明。我们还证明，从系统的$N$ δ中心删除一个单一的中心转移所有的束缚态能级向上作为一个简单的后果的柯西交错定理。
英文摘要：
We give a brief exposition of the formulation of the bound state problem for the one-dimensional system of $N$ attractive Dirac delta potentials, as an $N \times N$ matrix eigenvalue problem ($ΦA =ωA$). The main aim of this paper is to illustrate that the non-degeneracy theorem in one dimension breaks down for the equidistantly distributed Dirac delta potential, where the matrix $Φ$ becomes a special form of the circulant matrix. We then give an elementary proof that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of $N$ delta centers shifts all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem.
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