One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials

TL;DR

This paper analyzes the spectral and scattering properties of a one-dimensional semirelativistic quantum system with multiple Dirac delta potentials, providing explicit formulas, numerical results, and insights into bound states, scattering, and renormalization.

Contribution

It introduces a resolvent formula using a principal matrix for the semirelativistic Hamiltonian with multiple delta potentials, and thoroughly investigates bound states, scattering, and renormalization effects.

Findings

Maximum of N bound states for N potentials

Explicit bound state wave functions derived

Reflection and transmission coefficients computed

Abstract

In this paper, we consider the one-dimensional semirelativistic Schr\"{o}dinger equation for a particle interacting with $N$ Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an $N \times N$ 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 self-adjoint 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.