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We give a brief exposition of the formulation of the bound state problem forthe 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 thispaper is to illustrate that the non-degeneracy theorem in one dimension breaksdown for the equidistantly distributed Dirac delta potential, where the matrix$\Phi$ becomes a special form of the circulant matrix. We then give anelementary proof that the ground state is always non-degenerate and theassociated wave function may be chosen to be positive by using thePerron-Frobenius theorem. We also prove that removing a single center from thesystem of $N$ delta centers shifts all the bound state energy levels upward asa simple consequence of the Cauchy interlacing theorem.

A Singular One-Dimensional Bound State Problem and its Degeneracies