Delta Potential Research Articles

Infinite bound states and hydrogen atom-like energy spectrum induced by a flat band

In this work, we investigate the bound-state problem in a one-dimensional spin-1 Dirac Hamiltonian with a flat band. It is found that the flat band has significant effects on the bound states. For example, for Dirac delta potential gδ(x), there exists one bound state for both the positive and negative potential strength g. Furthermore, when the potential is weak, the bound-state energy is proportional to the potential strength g. For square well potential, the flat band results in the existence of infinite bound states for arbitrarily weak potential. In addition, when the bound-state energy is very near the flat band, the energy displays a hydrogen atom-like spectrum, i.e. the bound-state energies are inversely proportional to the square of the natural number n (e.g., E n ∝ 1/n 2, n = 1, 2, 3, …). Most of the above nontrivial behaviors can be attributed to the infinitely large density of states of the flat band and its ensuing 1/z singularity of the Green function. The combination of a short-ranged potential and flat band provides a new possibility to get an infinite number of bound states and a hydrogen atom-like energy spectrum. In addition, our findings provide some useful insights and further our understanding of the many-body physics of the flat band.

Scattering state study of fermions due to q-deformed Dirac delta potential

In this paper, by introducing the q-deformed delta function, the Dirac equation for a fermion in the presence of Dirac delta scalar and vector potential is developed. Then, considering the boundary condition and the effect of the q-deformed Dirac delta potential, the scattering state is investigated. Further, the reflection and transmission coefficients are determined. Moreover, the effect of the q-parameter and the potential coefficient on the reflection and transmission coefficients is reported.

Noninteracting fermionic systems with localized losses: Exact results in the hydrodynamic limit

We investigate the interplay between unitary dynamics after a quantum quench and localized dissipation in a noninteracting fermionic chain. In particular, we consider the effect of gain and loss processes, for which fermions are added and removed incoherently. We focus on the hydrodynamic limit of large distances from the source of dissipation and of long times, with their ratio being fixed. In this limit, the localized dissipation gives rise to an effective delta potential (dissipative impurity), and the time-evolution of the local correlation functions admits a simple hydrodynamic description in terms of the fermionic occupations in the initial state and the reflection and transmission amplitudes of the impurity. We derive this hydrodynamic framework from the ab initio calculation of the microscopic dynamics. This allows us to analytically characterize the effect of dissipation for several theoretically relevant initial states, such as a uniform Fermi sea, homogeneous product states, or the inhomogeneous state obtained by joining two Fermi seas. In this latter setting, when both gain and loss processes are present, we observe the emergence of exotic nonequilibrium steady states with stepwise uniform density profiles. In all instances, for strong dissipation the coherent dynamics of the system is arrested, which is a manifestation of the celebrated quantum Zeno effect.

Dispersive dromions, conserved densities and fluxes with integrability via P-test for couple of nonlinear dynamical system

The famous Hamiltonian amplitude equation (HAE) and the well known cubic nonlinear Schrödinger equation (CNLSE) with repulsive delta potential have been taken into account for integrability analysis, establishment of conservation laws (CLs) and traveling wave (soliton) solutions in the polynomial forms. The integrability investigation has been done with the help of Painlevé test (P-test), conserved densities and associated fluxes required for the establishment of CLs are found with the help of scaling invariance approach whereas the traveling wave solutions in the form of polynomials are derived by unified method (UM).

Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potentiali∂tu+∂x2u+γδ(x)u+|u|p−1u=0,(t,x)∈R×R, at the mass-energy threshold, namely, when Eγ(u0)[M(u0)]σ=E0(Q)[M(Q)]σ where u0∈H1(R) is the initial data, Q is the ground state of the free NLS on the real line R, Eγ is the energy, M is the mass and σ=(p+3)/(p−5). We also prove failure of the uniform space-time bounds at the mass-energy threshold.

Modified Dirac delta function and modified dirac delta potential in the quantum mechanics

Modified Dirac delta function and modified dirac delta potential in the quantum mechanics

Soliton solutions and periodic solutions for two models arises in mathematical physics

<abstract><p>We aimed in this paper to acquire the periodic wave solutions and soliton solutions and other solutions such as kink-wave solutions for the cubic nonlinear Schrödinger equation with repulsive delta potential ($ \delta $-NLSE) and complex coupled Higgs field equation via two mathematical methods Jacobi elliptic function method and generalized Kudryashov method. Some of these solutions are degenerated to solitary wave solutions and periodic wave solutions in the limit case. We also gave the meaning of these solutions physically and the numerical simulation by some figures.</p></abstract>

Instability of the solitary waves for the 1d NLS with an attractive delta potential in the degenerate case

Instability of the solitary waves for the 1d NLS with an attractive delta potential in the degenerate case

Variational approach to the Schrödinger equation with a delta-function potential

We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Thermo-Magneto-Electric Transport through a Torsion Dislocation in a Type I Weyl Semimetal

Herein, we study electronic and thermoelectric transport in a type I Weyl semimetal nanojunction, with a torsional dislocation defect, in the presence of an external magnetic field parallel to the dislocation axis. The defect is modeled in a cylindrical geometry, as a combination of a gauge field accounting for torsional strain and a delta-potential barrier for the lattice mismatch effect. In the Landauer formalism, we find that due to the combination of strain and magnetic field, the electric current exhibits chiral valley-polarization, and the conductance displays the signature of Landau levels. We also compute the thermal transport coefficients, where a high thermopower and a large figure of merit are predicted for the junction.

Propagator calculations for time dependent Dirac delta potentials and corresponding two state models

Time dependent Dirac delta potential is an interesting problem in quantum mechanics due to its applications in various fields. In this paper we have worked on this problem for two different models. We have derived the propagator for scattering problems in presence of a Dirac delta potential. Here strength of the potential is a function of time i.e.f(t). We have devised a mathematical method to solve this problem using Green's function method. We have also solved the corresponding two state model and find exact results. Our results are purely analytical and we have reported exact expressions of ψ(x′,t) in all cases.

Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions

We offer a consistent dynamical formulation of stationary scattering in two and three dimensions that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a $2\times 2$ matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. In particular, we show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices each related to the time-evolution operator for an effective non-unitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in two and three dimensions and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.

Vector Soliton in Coupled Nonlinear Schrödinger Equation

Researchers are currently interested in studying the dynamics of the wave field in a nonlinear and dispersive medium. The Nonlinear Schrödinger Equation (NLSE), which is the fundamental equation that explains the phenomenon, has paved the way for research in a variety of fields, including soliton scattering. However, if the fields have a large number of components, the Coupled NLSE should be considered. We used orthogonally polarised and equal-amplitude vector solitons with two polarization directions to model the interactions. The effect of vector soliton scattering by external Delta potential in Coupled NLSE was studied in this paper. The scattering process is primarily determined by the initial velocity, amplitude of the soliton and potential strength. The variational approximation and direct numerical methods of Coupled NLSE were used to investigate the scattering process. The variational approximation (VA) method was used to analyse the dynamics of soliton’s width and center of mass position. The soliton may thus be reflected, transmitted or trapped within the potential. Uncoupled solitons may initially create a coupled state if their kinetic energy is less than the attractive interaction potential between solitons, but once their velocity surpasses the critical velocity, the soliton will easily pass through each other. To validate the approximation, a direct numerical simulation of CNLSE was performed. The results of the VA method and direct numerical simulation of Coupled NLSE are in good agreement when the parameters for both solutions are set to the same value. The initial velocity, potential strength and soliton amplitude play a role in the scattering of the vector soliton with Delta potential.

Exact results for the Schrödinger equation with moving localized potential

We have provided an alternative method to solve the time-dependent Schrödinger equation for a moving delta function potential with a time-varying strength. We have employed the Galilean transformation to get the equation corresponding to a static delta potential with a time-varying strength. We apply the Feynman's operator technique along with the Green's function method to solve the equation. One of the observation is if a particle initially in the non eigenstate, can posses eigen state after encountered by a attractive potential well in the same initial location. By applying our proposed method, we have derived the analytical time domain solution for a constant and time inversely decaying strength of a uniformly moving well. We observed that the particle is unlikely to be found in the moving well for this decaying strength at large times unlike the previously observed in the case of constant strength.

The variational method applied to the harmonic oscillator in the presence of a delta function potential

The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension, where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence, to gain more insight, one can attempt to use model wave functions which are explicitly and simply expressed. Here, we apply the variational method to verify how closely one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies, which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.

Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions

A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection (R) and the transmission (T) coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth–Uhlenbeck formula, and we make comments about the proper ϵB→0 (where ϵB is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan’s contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.

Impurities in systems of noninteracting trapped fermions

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

Vacuum Polarization with Zero-Range Potentials on a Hyperplane

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension 1 in (d+1)-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviors of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are softened in the specific case of a pure Dirac delta potential.

An analytical solution for quantum scattering through a $${\cal P}{\cal T}$$-symmetric delta potential

We employ the Lippmann-Schwinger formalism to derive the analytical solutions of the transmission and reflection coefficients through a one-dimensional open quantum system, in which particle loss or gain on one lattice site located at x = 0, or particle loss and gain on the lattice sites located at $$x = \pm {\textstyle{L \over 2}}$$ are considered respectively. The gain and loss on the lattice site are modeled by the delta potential with positive and negative imaginary values. The analytical solution reveals the underlying physics that the sum of the transmission and reflection coefficients through an open quantum system (even a $${\cal P}{\cal T}$$ -symmetric open system) may not be 1, i.e., qualitatively explains that the number of particles is not conserved in an open quantum system. Furthermore, we find that the resonance states can be formed in the $${\cal P}{\cal T}$$ -symmetric delta potential, which is similar to the case of real delta potential. The results of our analysis can be treated as the starting point of studying quantum transport problems through a non-Hermitian system using Green’s function method, and more general cases for high-dimensional systems may be deduced by the same procedure.

Effective action for delta potentials: Spacetime-dependent inhomogeneities and Casimir self-energy

We study the vacuum fluctuations of a quantum scalar field in the presence of a thin and inhomogeneous flat mirror, modeled with a delta potential. Using heat-kernel techniques, we evaluate the Euclidean effective action perturbatively in the inhomogeneities (nonperturbatively in the constant background). We show that the divergences can be absorbed into a local counterterm and that the remaining finite part is in general a nonlocal functional of the inhomogeneities, which we compute explicitly for massless fields in $D=4$ dimensions. For time-independent inhomogeneities, the effective action gives the Casimir self-energy for a partially transmitting mirror. For time-dependent inhomogeneities, the Wick-rotated effective action gives the probability of particle creation due to the dynamical Casimir effect.