Transmission resonances and supercritical states in a one-dimensional cusp potential (original) (raw)

J Phys a Math Gen, 2000

An elementary treatment of the Dirac Equation in the presence of a threedimensional spherically symmetric δ(r − r 0)-potential is presented. We show how to handle the matching conditions in the configuration space, and discuss the occurrence of supercritical effects.

Reflectionless PT-symmetric potentials in the one-dimensional Dirac equation

Journal of Physics A, 2010

We study the one-dimensional Dirac equation with local PT -symmetric potentials whose discrete eigenfunctions and continuum asymptotic eigenfunctions are eigenfunctions of the PT operator, too: on these conditions the bound-state spectra are real and the potentials are reflectionless and conserve unitarity in the scattering process. Absence of reflection makes it meaningful to consider also PT -symmetric potentials that do not vanish asymptotically.

On scattering from the one-dimensional multiple Dirac delta potentials

European Journal of Physics

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 arbitrary number of centers and study threshold anomaly for N = 2 and N = 4 cases. We also study further features like quantum metastable states like resonances, including their corresponding Gamow functions, and virtual or antibound states. The use of the Lippmann-Schwinger equation simplifies enormously our analysis and gives exact results for an arbitrary number of Dirac delta potential.

Non-local PT-symmetric potentials in the one-dimensional Dirac equation

Journal of Physics A, 2008

The Dirac equation in (1+1) dimensions with a non-local PTsymmetric potential of separable type is studied by means of the Green function method: properties of bound and scattering states are derived in full detail and numerical results are shown for a potential kernel of Yamaguchi type, inspired by the treatment of low-energy nucleonnucleon interaction.

The Dirac equation with a δ-potential

Journal of Physics A: Mathematical and General, 2000

An elementary treatment of the Dirac Equation in the presence of a threedimensional spherically symmetric δ(r − r 0 )-potential is presented. We show how to handle the matching conditions in the configuration space, and discuss the occurrence of supercritical effects.

Reflectionless {\cal P}{\cal T} -symmetric potentials in the one-dimensional Dirac equation

Journal of Physics A: Mathematical and Theoretical, 2010

Revisiting double Dirac delta potential

European Journal of Physics, 2016

We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission (T = 1). Perfect transmission energies turn out to be the critical property of symmetric and antisymmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, R(E)] at energy E = 0 is non-intuitive. However, earlier this has been found and called "threshold anomaly". Here we show that it is a critical phenomenon and we can have 0 ≤ R(0) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate (ψ(x) = Ax + B) of delta well between two symmetric rigid walls for R(0) = 0. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.

Double-Delta Potentials: One Dimensional Scattering

International Journal of Theoretical Physics, 2011

The path is explored between one-dimensional scattering through Dirac-δ walls and one-dimensional quantum field theories defined on a finite length interval with Dirichlet boundary conditions. It is found that two δ's are related to the Casimir effect whereas two δ's plus the first transparent Pösch-Teller well arise in the context of the sine-Gordon kink fluctuations, both phenomena subjected to Dirichlet boundary conditions. One or two delta wells will be also explored in order to describe absorbent plates, even though the wells lead to non unitary Quantum Field Theories.

One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

Physical Review D, 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 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.

Bound-state solutions of the Dirac-Rosen-Morse potential with spin and pseudospin symmetry⋆

The European Physical Journal A, 2010

The energy spectra and the corresponding two-component spinor wavefunctions of the Dirac equation for the Rosen-Morse potential with spin and pseudospin symmetry are obtained. The s−wave (κ = 0 state) solutions for this problem are obtained by using the basic concept of the supersymmetric quantum mechanics approach and function analysis (standard approach) in the calculations. Under the spin symmetry and pseudospin symmetry, the energy equation and the corresponding twocomponent spinor wavefunctions for this potential and other special types of this potential are obtained. Extension of this result to κ = 0 state is suggested.