A singular one-dimensional bound state problem and its degeneracies

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 ×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 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.

On scattering from the one-dimensional multiple Dirac delta potentials

“…This is due to the appearance of a second bound state at this critical value. Note that this is exactly the same condition for the formation of the second bound state, as discussed in our recent paper [31] (the only difference is that the centers are separated by a in there). In other words, the underlying reason of this is the appearance of a bound state very close to the threshold energy [7].…”

“…The bound state energies can be found from the non-trivial solution of the matrix equation for Φ, that is, the bound state energies must satisfy det Φ(E) = 0. This was explicitly shown in our recent paper [31]. Moreover, one can also imagine the following eigenvalue problem for the matrix Φ, i.e.,…”

“…The simple poles on the positive imaginary k axis correspond to the bound states, whereas the ones in the negative imaginary k axis are known as virtual states [20]. As emphasized in [31], it is well known that there are two bound states when a is sufficiently large (a > 2 /mλ or the critical case a = 2 for particular choice of the parameter λ = 2) and the second bound state eventually becomes a virtual state as we decrease the distance between the centers. From Fig.…”

An approximation to the Woods–Saxon potential based on a contact interaction

“…In spite of their simplicity, they have a vast amount of applications in modelling real physical systems, as we can see for instance in a recent review [1], some books [2,3] and the references therein. The landmark example in solid state physics is a limiting case of the Kronig-Penney model in which a countably infinite set of Dirac delta interactions are periodically distributed along a straight line [4][5][6]. Other examples of physical interest are the following: a Bose-Einstein condena e-mail: manuelgadella1@gmail.com (corresponding author) 0123456789().…”

A Perturbative Approach to the Tunneling Phenomena

“…The same method can also be applied to the one-dimensional case. In the symmetrically placed Dirac delta potentials with equal strengths λ, the exact bound state energies when they are sufficiently far away from each other (when a > 1/λ, there are two bound state energies) can analytically be computed [42]…”