Delta Potential Research Articles

Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential

We consider the NLS equation with a linear double-well potential. Symmetry breaking, i.e. the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken, i.e. unfolded, and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram.

Bound states of spin–orbit coupled cold atoms in a Dirac delta-function potential

Dirac delta-function potential is widely studied in quantum mechanics because it usually can be exactly solved and at the same time is useful in modeling various physical systems. Here we study a system of delta-potential trapped spin–orbit coupled cold atoms. The spin–orbit coupled atomic matter wave has two kinds of evanescent modes, one of which has a pure imaginary wavevector and is an ordinary evanescent wave; while the other with a complex number wave vector is recognized as an oscillating evanescent wave. We identified the eigenenergy spectra and the existence of bound states in this system. The bound states can be constructed analytically using the two kinds of evanescent modes and we found that they exhibit typical features of stripe phase, separated phase or zero-momentum phase. In addition to that, the properties of semi-bound states are also discussed, which is a localized wave packet on a plane wave background.

Dirichlet forms and polymer models based on stable processes

In this paper, we are concerned with polymer models based on α-stable processes, where α∈(d2,d∧2) and d stands for dimension. They are attached with a delta potential at the origin and the associated Gibbs measures are parametrized by a constant γ∈R∪{−∞} playing the role of inverse temperature. Phase transition exhibits with critical value γcr=0. Our first object is to formulate the associated Dirichlet form of the canonical Markov process X(γ) induced by the Gibbs measure for a globular state γ>0 or the critical state γ=0. Approach of Dirichlet forms also leads to deeper descriptions of their probabilistic counterparts. Furthermore, we will characterize the behaviour of polymer near the critical point from probabilistic viewpoint by showing that X(γ) is convergent to X(0) as γ↓0 in a certain meaning.

The Propagators for δ and δ′ Potentials With Time-Dependent Strengths

We study the time-dependent Schrodinger equation with nite number of Dirac \delta and\delta' potentials with time dependent strengths in one dimension. We obtain the formal solution for generic timedependent 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.

Threshold of discrete Schrödinger operators with delta potentials on n-dimensional lattice

ABSTRACT Eigenvalue behaviours of Schrödinger operator defined on n-dimensional lattice with n + 1 delta potentials are studied. It can be shown that lower threshold eigenvalue and lower threshold resonance appear for , and lower super-threshold resonance appears for n = 1.

Singular Mean-Field States: A Brief Review of Recent Results

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.

Nonlinear Schrödinger equation with a Dirac delta potential: finite difference method**Project supported by the National Natural Science Foundation of China (Grant Nos. 11601517, 61772542).

The nonlinear Schrödinger equation with a Dirac delta potential is considered in this paper. It is noted that the equation can be transformed into an equation with a drift-admitting jump. Then following the procedure proposed in Chen and Deng (2018 Phys. Rev. E 98 033302), a new second-order finite difference scheme is developed, which is justified by numerical examples.

On finite temperature Casimir effect for Dirac lattices

We consider polarizable sheets modeled by a lattice of delta function potentials. The Casimir interaction of two such lattices is calculated at nonzero temperature. The heat kernel expansion for periodic singular background is discussed in relation with the high temperature asymptote of the free energy.

Small Scale Equidistribution for a Point Scatterer on the Torus

We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two- and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the "new" eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the "new" eigenfunctions at certain scales.

Green's function formulation of multiple nonlinear Dirac δ-function potential in one dimension

In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants.

Non-additive quantum mechanics for a position-dependent mass system: Dirac delta and quasi-periodic potentials

Motivated by non-extensive statistical mechanics, in this work we consider a deformed Schrödinger equation (DSE) for position-dependent mass (PDM) systems, whose deformed plane-wave solutions allow to characterise a non-periodic lattice. We obtain a deformed version of the Bloch theorem and we illustrate the formalism presented with two examples of the literature: the Dirac and the Kronig-Penney potentials. We found that the Kronig-Penney potential offers a modelling for a lattice with defects expressed by a non-periodicity of the potential within the underlying non-extensive mathematical structure, which is evidenced by the displacement of the gaps with respect to the non-deformed case. The eigenfunctions, the reduced energy bands scheme and the density of states are affected by the deformation.

Analysis of the Breakdown of Exponential Decays of Resonances

In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schr\odinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.

Thermal properties of magnetopolaron in a GaAs delta potential under Rashba effect

Abstract This work treats the polaron's magnetic and thermal properties with delta confinement and interacting with external magnetic and electric areas have been investigated under Rashba effect. We used the Schrodinger equation to find the wave function and the quantified energies of the system. We found that the corresponding wave function can be separated into the product of the longitudinal and transverse gaussian wave function. Moreover, we have obtained an almost exact expression of the system's power spectrum in terms of Rashba spin orbit coupling, delta parameters and electromagnetic field. We have calculated the most common magnetic and thermal elements namely the free energy, entropy, the magnetization, heat capacity and the magnetic susceptibility with the help of the partition function. We found that the results for magnetic and thermal features are strongly affected by external fields, delta length and temperature. Moreover, the results show that the system can cross from ferromagnetic to paramagnetic phase by adjusting magnetic properties.

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ -function potential

In contrast with the 3D result, the Beth-Uhlenbeck (BU) formula in 1D contains an extra -1/2 term. The origin of this -1/2 term is explained using a spectral density approach. To be explicit, a delta-function potential is used to show that the correction term arises from a pole of the density of states at zero energy. The spectral density method shows that this term is actually an artifact of the non-normalizability of the scattering states and an infrared cutoff regularization scheme has to be used to get the correct result in 1D. The formal derivation of the BU formula would miss this term since it ignores the effects of the boundary terms. While the result is shown for the delta-function potential, the method and result are valid for more general potentials. Additionally, the 1D Levinson's theorem can be extracted from the spectral density method using the asymptotic form of general potentials. The importance of the result lies in the fact that all these correction terms in 1D have a universal source: a pole at zero energy. Similar calculations using quantum field theoretical approaches (without explicit infrared cutoff regularization schemes) also show the same subtleties with the correction term originating from the zero energy scattering states (appendix A).

Nonlocality amplifies echoes

In this paper we will provide smoking-gun signatures of nonlocal interactions while studying reflection and transmission of waves bouncing through two Dirac delta potentials. In particular, we will show that the transmission of waves is less damped compared to the local case, due to the fact that nonlocality weakens the interaction. As a consequence the echoes are amplified. These signatures can be potentially detectable in the context of gravitational waves, where two Dirac delta potentials can mimic the two potential barriers at the surface and at the photon sphere of an ultra compact object, or, at the two photon spheres of a wormhole, experiencing nonlocal interactions.

Shannon information entropy for a one-dimensional ionic crystal

In this work, we present the information theoretic lengths given by Shannon and Fisher for the lowest-lying stationary state of a one-dimensional ionic crystal modeled by [Formula: see text] equally spaced attractive and repulsive Dirac delta potentials. The entropic uncertainty relations related to position and momentum spaces are studied as a function of the number of ions and the distance between them. Our results show that the stability of the ionic crystal depends on the number of ions and distance between them. In particular, we show that the position Shannon entropy is always positive and increases as the lattice constant between ions grows, in contrast with the momentum Shannon entropy which decreases and becomes negative beyond a particular lattice constant value.

Spintronics – A mini review

Abstract Over the last decade, the construction and analysis of spintronic devices have been explored. This mini-review focuses on the basic spintronic phenomena such as giant magnetoresistance, tunnel magnetoresistance, spin transfer torque, spin Hall effect, spin Seebeck effect and BGKC (Bruno – Gnanasekar – Karunakaran – Chandramohan) observation. The mechanism of spin-orbit interaction is theoretically discussed. The in-plane wave vector, magnetic field, bias voltage and the delta potential are influenced on spin-polarized transport in heterostructures. The comparison between the role of Dresselhaus and Rashba spin-orbit interaction in the heterostructure is investigated. The principle of spintronic devices (sensors, magnetic random access memory, unipolar spin diode, spin photodiode, spin LED, spin solar cell, spin FET) and spintronic instruments (spin SEM, spin STM) are also explained.

Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential

We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schrodinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a frequency, and $\phi_\omega\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrodinger equations with a harmonic potential, Ohta~(2018) proved that if $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, then the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrodinger equations with other potentials such as an attractive Dirac delta potential.

Applications of the worldline Monte Carlo formalism in quantum mechanics

In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this “worldline Monte Carlo” approach to the standard problem of the numerical approximation of the non-relativistic path integral, resulting in a formalism whose characteristic feature is the fast, non-recursive generation of an ensemble of trajectories that is independent of the potential, and is thus universally applicable. The numerical implementation discretises the trajectories with respect to their time parametrisation but maintains a continuous spatial domain. In the case of singular potentials, the discretised action gets adapted to the singularity through a “smoothing” procedure. We show for a variety of examples (the harmonic oscillator in various dimensions, the modified Pöschl–Teller potential, delta-function potentials, the Coulomb and Yukawa potentials) that the method allows one to obtain fast and reliable estimates for the Euclidean propagator and use them in a certain time window suitable for extracting the ground state energy. Usually Monte Carlo calculations in quantum mechanics are performed using algorithms that involve a more specific adaptation to the potential, and we briefly compare our method with the most widely used such algorithms, which are of Metropolis type. As an aside, we apply it for studying the classical limit where nearly classical trajectories are expected to dominate in the path integral. We expect the advances made here to be useful also in the relativistic case.

Instability of standing waves for nonlinear Schrödinger equation with delta potential

We consider nonlinear Schrodinger equation with an attractive delta potential in one space dimension. We review some recent results on relations between orbitally unstable standing wave solutions and finite time blowup solutions.