One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

Abstract

In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with N Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an N × N matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most N bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a selfadjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for N centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.

A Perturbative Approach to the Tunneling Phenomena

“…Furthermore, point like Dirac delta interactions have also been extended to various more general cases. For our approach, to illustrate the main ideas, we are mainly concerned with the delta potentials supported by points on flat and hyperbolic manifolds [13,14,15], and delta potentials supported by curves in flat spaces, and its various relativistic extensions in flat spaces [16,17,18,19].…”

Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity

“…They are also used in the study of heterostructures, where they may appear in connection with an abrupt effective mass change [5]. In addition, they constitute a class of solvable or quasi-solvable potentials suitable to study basic quantum properties, stationary states, scattering, resonances, etc, where one uses point interactions of the form δ or δ ′ or a linear combination thereof [6][7][8][9][10][11][12][13][14][15][16][17]. A rigorous definition of perturbations of the form −a δ(x) + b δ ′ (x), a and b being real numbers, is rather technical and we are not going to deal with this here.…”

“…We first introduce an ultraviolet energy cut-off and then the ensuing renormalisation of the coupling constant. Renormalisation procedures are necessary for point potentials not only in higher dimensions [1,2], but also in one-dimensional problems when the kinetic energy operator is proportional to the magnitude of the momentum and not to its square, as in the relativistic case [6][7][8]. Again, eigenvalues are obtained through a transcendental equation involving the Airy function and its derivative.…”

Spectral properties of the two-dimensional Schrödinger Hamiltonian with various solvable confinements in the presence of a central point perturbation

“…Spectral properties of the two-dimensional Schrödinger Hamiltonian with various solvable confinements in the presence of a central point perturbation S Fassari 1,2,3 , M Gadella 1,4,6 , M L Glasser 1,5 , L M Nieto 1,4 and F Rinaldi 2,3…”