Revisiting double Dirac delta potential

Ahmed, Zafar; Kumar, Sachin; Sharma, Mayank; Sharma, Vibhu Saujanya

doi:10.1088/0143-0807/37/4/045406

Cited by 18 publications

(37 citation statements)

References 14 publications

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“…(15,16) represent the negative energy physical poles of r(E). The bound state eigenvalues are the common poles of the reflection and transmission amplitudes [15] of a one-dimensional potential well. From the solutions (15,16) we can identify the zero-energy HBS as of odd and even parity ψ * (x) = A sgn(x)J 0 (q exp(−|x|/a)), when ψ * (0) = J 0 (q) = 0, and…”

Section: B the Exponential Potential Wellmentioning

confidence: 99%

The paradoxical zero reflection at zero energy

Ahmed¹,

Sharma²,

Sharma³

et al. 2016

Eur. J. Phys.

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Usually, the reflection probability R(E) of a particle of zero energy incident on a potential which converges to zero asymptotically is found to be 1: R(0) = 1. But earlier, a paradoxical phenomenon of zero reflection at zero energy (R(0) = 0) has been revealed as a threshold anomaly. Extending the concept of Half Bound State (HBS) of 3D, here we show that in 1D when a symmetric (asymmetric) attractive potential well possesses a zero-energy HBS, R(0) = 0 (R(0) << 1). This can happen only at some critical values q c of an effective parameter q of the potential well in the limit E → 0 + . We demonstrate this critical phenomenon in two simple analytically solvable models which are square and exponential wells. However, in numerical calculations even for these two models R(0) = 0 is observed only as extrapolation to zero energy from low energies, close to a precise critical value q c . By numerical investigation of a variety of potential wells, we conclude that for a given potential well (symmetric or asymmetric), we can adjust the effective parameter q to have a low reflection at a low energy. * Electronic address: 1:zahmed@barc.gov.in, 2:

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Section: B the Exponential Potential Wellmentioning

confidence: 99%

The paradoxical zero reflection at zero energy

Ahmed¹,

Sharma²,

Sharma³

et al. 2016

Eur. J. Phys.

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“…We even plotted figure 1(c) in [2] showing the zero-energy, piece-wise zero-curvature, one-node half-bound state under this condition (equation (6) of the Comment). In our earlier paper [3], we already gave a general expression for r ( ) E (see equation (11) in [3]) of the double delta potential where the distance between the delta functions is a (not a 2 ). The three most simple expressions of R(0) are given with respect to the condition that energy in a well-barrier system: ll < 0.…”

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confidence: 99%

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Ahmed

Sharma

et al. 2017

Eur. J. Phys.

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“…For E 1 B = E 2 B = E B and a 1 = −a 2 = −a, the behavior of the reflection coefficient as a function of ka is shown below for particular values of a|E B |. This is also a typical behavior of the reflection coefficient in the nonrelativistic case [46,47]. The particle is fully transmitted at some certain energies that can be seen easily from Fig.…”

Section: The Bound States and The Scattering Problem In The Masslementioning

confidence: 69%

“…For example, the transmission coefficient T (k) has some sharp peaks around certain values of k where it becomes unity. These peaks have been interpreted as resonances by some authors [45][46][47] in the nonrelativistic case. However, they should not be confused with resonances as unstable states in quantum mechanics [48].…”

Section: Introductionmentioning

confidence: 83%

One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

2017

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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.

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Revisiting double Dirac delta potential

Cited by 18 publications

References 14 publications

The paradoxical zero reflection at zero energy

The paradoxical zero reflection at zero energy

Reply to Comment on ‘The paradoxical zero reflection at zero energy’

One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

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