Abstract.
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 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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Erman, F., Gadella, M., Tunalı, S. et al. A singular one-dimensional bound state problem and its degeneracies. Eur. Phys. J. Plus 132, 352 (2017). https://doi.org/10.1140/epjp/i2017-11613-7
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DOI: https://doi.org/10.1140/epjp/i2017-11613-7