A Singular One-Dimensional Bound State Problem and its Degeneracies
| Autor: | Erman, F., Gadella, M., Tunalı, Ş., Uncu, H. |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | Eur. Phys. J. Plus 132, 352 (2017) |
| Druh dokumentu: | Working Paper |
| Popis: | 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 ($\Phi A =\omega A$). 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 $\Phi$ 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. Comment: Major modifications: title changed, typos corrected, clarifications added, published version |
| Databáze: | arXiv |
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