GCRIS

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Citation - WoS: 20
Citation - Scopus: 23
One-Dimensional Semirelativistic Hamiltonian With Multiple Dirac Delta Potentials
(American Physical Society, 2017) Erman, Fatih; Gadella, Manuel; Uncu, Haydar
In this paper, we consider the one-dimensional semirelativistic Schrdinger 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 x 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.
Citation - WoS: 5
Citation - Scopus: 5
The Propagators for Δ and Δ′ Potentials With Time-Dependent Strengths
(Frontiers Media S.A., 2020) Erman, Fatih; Gadella, Manuel; Uncu, Haydar
We study the time-dependent Schrodinger equation with finite number of Dirac delta and delta ' potentials with time dependent strengths in one dimension. We obtain the formal solution for generic time dependent strengths and then we study the particular cases for single delta potential and limiting cases for finitely many delta potentials. Finally, we investigate the solution of time dependent Schrodinger equation for delta ' potential with particular forms of the strengths.
Citation - WoS: 16
Citation - Scopus: 15
On Scattering From the One-Dimensional Multiple Dirac Delta Potentials
(Institute of Physics Publishing, 2018) Erman, Fatih; Gadella, Manuel; Uncu, Haydar
In this paper, we propose a pedagogical presentation of the Lippmann-Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann-Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann-Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.
Citation - WoS: 16
Citation - Scopus: 16
A Singular One-Dimensional Bound State Problem and Its Degeneracies
(Springer Verlag, 2017) Erman, Fatih; Gadella, Manuel; Tunalı, Seçil; Uncu, Haydar
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.

Mathematics / Matematik

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