# A Singular One-Dimensional Bound State Problem and its Degeneracies

## TL;DR

This paper analyzes the bound state problem for a one-dimensional system with multiple attractive Dirac delta potentials, revealing degeneracy conditions, proving the non-degeneracy of the ground state, and examining effects of removing potential centers.

## Contribution

It demonstrates that the non-degeneracy theorem fails for equidistant delta potentials and provides elementary proofs for ground state properties and energy shifts upon removing centers.

## Key findings

- Degeneracy occurs in equidistant Dirac delta arrangements due to circulant matrix structure.
- Ground state is always non-degenerate and positive, proven via Perron-Frobenius theorem.
- Removing a delta center raises all bound state energies, shown by Cauchy interlacing theorem.

## 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 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.