One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
PHYSICAL REVIEW D 95, 045004 (2017) One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials Fatih Erman* Department of Mathematics, İzmir Institute of Technology, Urla 35430, İzmir, Turkey † Manuel Gadella Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Campus Miguel Delibes, Paseo Belén 7, 47011 Valladolid, Spain ‡ Haydar Uncu Department of Physics, Adnan Menderes University, 09100 Aydın, Turkey (Received 18 August 2016; published 17 February 2017) 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 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.
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