Δ-function Potentials Research Articles

Scattering by a collection of [formula omitted]-function point and parallel line defects in two dimensions

Interaction of waves with point and line defects is usually described by δ-function potentials supported on points or lines. In two dimensions, the scattering problem for a finite collection of point defects or parallel line defects is exactly solvable. This is not true when both point and parallel line defects are present. We offer a detailed treatment of the scattering problem for finite collections of point and parallel line defects in two dimensions. In particular, we perform the necessary renormalization of the coupling constants of the point defects, introduce an approximation scheme which allows for an analytic calculation of the scattering amplitude and Green’s function for the corresponding singular potential, investigate the consequences of perturbing this potential, and comment on the application of our results in the study of the geometric scattering of a particle moving on a curved surface containing point and line defects. Our results provide a basic framework for the study of spectral singularities and the corresponding lasing and antilasing phenomena in effectively two-dimensional optical systems involving lossy and/or active thin wires and parallel thin plates.

Quasinormal modes, echoes and the causal structure of the Green's function

Quasinormal modes describe the return to equilibrium of a perturbed system, in particular the ringdown phase of a black hole merger. But as globally-defined quantities, the quasinormal spectrum can be highly sensitive to global structure, including distant small perturbations to the potential. In what sense are quasinormal modes a property of the resulting black hole? We explore this question for the linearized perturbation equation with two potentials having disjoint bounded support. We give a composition law for the Wronskian that determines the quasinormal frequencies of the combined system. We show that over short time scales the evolution is governed by the quasinormal frequencies of the individual potentials, while the sensitivity to global structure can be understood in terms of echoes. We introduce an echo expansion of the Green's function and show that, as expected on general grounds, at any finite time causality limits the number of echoes that can contribute. We illustrate our results with the soluble example of a pair of δ-function potentials. We explicate the causal structure of the Green's function, demonstrating under what conditions two very different quasinormal spectra give rise to very similar ringdown waveforms.

Time-space noncommutativity and Casimir effect

We show that the Casimir force and energy are modified in the κ-deformed space-time. This is shown by solving the Green's function corresponding to κ-deformed scalar field equation in presence of two parallel plates, modelled by δ-function potentials. Exploiting the relation between Energy-Momentum tensor and Green's function, we calculate corrections to Casimir force, valid up to second order in the deformation parameter. The Casimir force is shown to get corrections which scale as L−4 and L−6 and both these types of corrections produce attractive forces. Using the measured value of Casimir force, we show that the deformation parameter should be below 10−23 m.

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

We use the transfer matrix formulation of scattering theory in two-dimensions (2D) to treat the scattering problem for a potential of the form where , and b are constants, is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of : (i) a linear combination of δ functions, in which case is a finite linear array of 2D δ functions; (ii) a linear combination of with real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and , the potentials and have the same scattering amplitude for .

Touching points in the energy band structure of bilayer graphene superlattices

The energy band structure of the bilayer graphene superlattices with zero-averaged periodic δ-function potentials are studied within the four-band continuum model. Using the transfer matrix method, the study is mainly focused on examining the touching points between adjacent minibands. For the zero-energy touching points the dispersion relation derived shows a Dirac-like double-cone shape with the group velocity which is periodic in the potential strength P with the period of π and becomes anisotropic at relatively large P. From the finite-energy touching points we have identified those located at zero wave-number. It was shown that for these finite-energy touching points the dispersion is direction-dependent in the sense that it is linear or parabolic in the direction parallel or perpendicular to the superlattice direction, respectively. We have also calculated the density of states and the conductivity which demonstrates a manifestation of the touching points examined.

RECURRENCE IN RESONANT TRANSMISSION OF ONE-DIMENSIONAL ARRAY OF DELTA POTENTIALS

In this paper, the resonant transmission of a moving particle which interacts with a one-dimensional array of N δ-function potentials is investigated. A suitable transfer matrix formulation is used to obtain the particle transmission. We give the parameters for perfect tunneling and the transcendental equation for the quasi-bound state energies for N = 2, 3 and 4. Conditions for perfect tunneling and resonant transmission are discussed for arrays with arbitrary N. A model to explain how the tunneling energy filter works in these systems is proposed here.

Split kinetic energy method for quantum systems with competing potentials

For quantum systems with competing potentials, the conventional perturbation theory often yields an asymptotic series and the subsequent numerical outcome becomes uncertain. To tackle such a kind of problems, we develop a general solution scheme based on a new energy dissection idea. Instead of dividing the potential energy into “unperturbed” and “perturbed” terms, a partition of the kinetic energy is performed. By distributing the kinetic energy term in part into each individual potential, the Hamiltonian can be expressed as the sum of the subsystem Hamiltonians with respective competing potentials. The total wavefunction is expanded by using a linear combination of the basis sets of respective subsystem Hamiltonians. We first illustrate the solution procedure using a simple system consisting of a particle under the action of double δ-function potentials. Next, this method is applied to the prototype systems of a charged harmonic oscillator in strong magnetic field and the hydrogen molecule ion. Compared with the usual perturbation approach, this new scheme converges much faster to the exact solutions for both eigenvalues and eigenfunctions. When properly extended, this new solution scheme can be very useful for dealing with strongly coupling quantum systems.

Phase modulation of charge density wave around a short-range potential by the Aharonov–Bohm effect

The analytic result of the Aharonov–Bohm effect for its influence on the oscillation of a two-dimensional charge density around a short range potential is given. Without losing generality, the cases of δ-function and hard disc potentials were examined. Numerical calculations show that the interferences among quantum particles are greatly influenced by the nonlocal effect, which leads to the modulation of phase and amplitude of the oscillation of charge density. Since the presence of a nonlocal influence of the Aharonov–Bohm effect on charged particles is universal, the results in the specific potentials examined are expected to appear also in other general systems which may be beneficial to the study of nanotechnology.

Layer-heterogeneous magnetic states in metallic nanosystems

The formation of layer-heterogeneous periodic magnetic states in metallic systems is explained in terms of the dependence of the energy of itinerant electrons on the magnetic ordering of localized spins. The interaction of the local magnetic moments with conduction electrons with a certain spin projection is described by a set of spin-dependent δ-function potentials. A matrix method is developed permitting one to calculate the energy spectrum and the density of states of itinerant electrons in the presence of a layered magnetic heterogeneity. This method is used to explain oscillations of the interlayer exchange interaction in metallic magnetic superlattices and the stabilization of spin-density wave structures in transition metals and alloys.

Calderón's problem for Lipschitz piecewise smooth conductivities

We consider Lipschitz conductivities which are piecewise smooth across polyhedral boundaries in . Using complex geometrical optics solutions for Schrödinger operators with certain δ-function potentials, we obtain global uniqueness for Calderón's inverse conductivity problem.

Gravitational and inertial mass of Casimir energy

It has been demonstrated, using variational methods, that quantum vacuum energy gravitates according to the equivalence principle, at least for the finite Casimir energies associated with perfectly conducting parallel plates. This conclusion holds independently of the orientation of the plates. We review these arguments and add further support to this conclusion by considering parallel semitransparent plates, that is, δ-function potentials, acting on a massless scalar field, in a spacetime defined by Rindler coordinates. We calculate the force on systems consisting of one or two such plates undergoing acceleration perpendicular to the plates. In the limit of small acceleration we recover (via the equivalence principle) the situation of weak gravity, and find that the gravitational force on the system is just Mg, where g is the gravitational acceleration and M is the total mass of the system, consisting of the mass of the plates renormalized by the Casimir energy of each plate separately, plus the energy of the Casimir interaction between the plates. This reproduces the previous result in the limit as the coupling to the δ-function potential approaches infinity. Extension of this latter work to arbitrary orientation of the plates, and to general compact quantum vacuum energy configurations, is under development.

Multiple scattering methods in Casimir calculations

Multiple scattering formulations have been employed for more than 30 years as a method of studying the quantum vacuum or Casimir interactions between distinct bodies. Here we review the method in the simple context of δ-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) After applying the method to rederive the Casimir force between two semitransparent plates and the Casimir self-stress on a semitransparent sphere, we obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. Simplifications occur for weak and strong coupling. In particular, after performing a power series expansion in the ratio of the radii of the objects to the separations between their centers, we are able to sum the weak-coupling expansions exactly to obtain explicit closed forms for the Casimir interaction energy. The same can be done for the interaction of a weak-coupling sphere or cylinder with a Dirichlet plane. We show that the proximity force approximation (PFA), which becomes the proximity force theorem when the objects are nearly touching each other, is very poor for finite separations.

How does Casimir energy fall? II. Gravitational acceleration of quantum vacuum energy

It has been demonstrated that quantum vacuum energy gravitates according to the equivalence principle, at least for the finite Casimir energies associated with perfectly conducting parallel plates. We here add further support to this conclusion by considering parallel semitransparent plates, that is, δ-function potentials, acting on a massless scalar field, in a spacetime defined by Rindler coordinates (τ, x, y, ξ). Fixed ξ in such a spacetime represents uniform acceleration. We calculate the force on systems consisting of one or two such plates at fixed values of ξ. In the limit of a large Rindler coordinate ξ (small acceleration), we recover (via the equivalence principle) the situation of weak gravity, and find that the gravitational force on the system is just Mg, where g is the gravitational acceleration and M is the total mass of the system, consisting of the mass of the plates renormalized by the Casimir energy of each plate separately, plus the energy of the Casimir interaction between the plates. This reproduces the previous result in the limit as the coupling to the δ-function potential approaches infinity.

Excitation and ionization of a simple two-level system by a harmonic force

A simple one-dimensional quantum system with two attractive δ-function potentials of strength q at x ± 1 is subjected to a spatially asymmetric (as in the dipole interaction) harmonic forcing with frequency ω. The time evolution of the system, which has two discrete energy levels −ωs < −ωa (depending on q) and a continuum spectrum, exhibits a rich dynamics including regimes where the rate of ionization becomes very small due to the 'inverse' Ramsauer effect in electron–atom collisions. The two-photon ionization with ω ≈ ωs/2 can be enhanced when ωa = ωs/2 though the one-photon ionization is not affected significantly by the location of excited level. When ω is very close to the one- or two-photon resonance the ionization rate can differ greatly from that given by the low order perturbation theory even for small forcing amplitude. This is caused in part by the fact that the dynamic Stark effect has a strong dependence on ω and may shift the resonance frequencies ωs, ωa up and down. When the ground state decays faster than the excited state and ω is not close to ωs − ωa, the excited level at late times becomes and remains more populated than the ground state. The occupations of the bound states oscillate with a frequency that can be quite low compared with ω, in particular in the case ω ≈ ωa ≈ ωs/2, but it approaches ω when ω ωs.Our analysis is based on the analytic structure of the wavefunction's Laplace transform in time. It considers one- and two-'photon' processes; the higher order multiphoton processes can also be treated by our computational scheme which goes beyond the low order perturbation theory.

Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems

A key element of theoretical physics is the conceptualisation of physical phenomena in terms of models, which are then investigated by the tools at hand. For quantum many-body systems, some models can be exactly solved, i.e., their physical properties can be calculated in an exact fashion. There is often a deep underlying reason why this can be done—the theory of integrability—which manifests itself in many guises. In Beautiful models, Bill Sutherland looks at exactly solved models in quantum many-body systems, a well established field dating back to Bethe's 1931 exact solution of the spin-1/2 Heisenberg chain. This field is enjoying a renaissance due to the ongoing and striking experimental advances in low-dimensional quantum physics, which includes the manufacture of quasi one-dimensional quantum gases. Apart from the intrinsic beauty of the subject material, Beautiful Models is written by a pioneering master of the field. Sutherland has aimed to provide a broad textbook style introduction to the subject for graduate students and interested non-experts. An important point here is the `language' of the book. In Sutherland's words, the subject of exactly solved models `belongs to the realm of mathematical physics—too mathematical to be `respectable' physics, yet not rigorous enough to be `real' mathematics. ...there are perennial attempts to translate this body of work into either respectable physics or real mathematics; this is not that sort of book.' Rather, Sutherland discusses the models and their solutions in terms of their `intrinisic' language, which is largely as found in the physics literature. The book begins with a helpful overview of the contents and then moves on to the foundation material, which is the chapter on integrability and non-diffraction. As is shown, these two concepts go hand in hand. The topics covered in later chapters include models with δ-function potentials, the Heisenberg spin chain, the Hubbard model, exchange models, the Calogero–Sutherland models and models with ground state wavefunctions of product form. One of my favourites is chapter 7, dealing with the consistency conditions for two-body scattering operators involving the Yang-Baxter equations. The strength of the book lies in the pedagogical approach, with the underlying emphasis on integrability and the Bethe ansatz. Sutherland brings his own insights to these problems, and as such there is also something to be gained by specialist readers. Given the author's aims, the book is not meant to be a complete and historic review of the field. Rather, the intention is that the general references given will point the reader in the right direction. Sutherland quite wisely gives an up-front apology to any authors he has left out. Certainly such historical omissions can be found. Two obvious examples are the origins of the condition of non-diffraction and twisted boundary conditions. Setting such minor points aside, Beautiful Models is a most welcome book. It does a great service to a fascinating, enduring and increasingly relevant field by highlighting not only the beauty, but also the magic of integrability.

Casimir energies and pressures for -function potentials

The Casimir energies and pressures for a massless scalar field associated with δ-function potentials in 1 + 1 and 3 + 1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1 + 1 dimension. The relation between Casimir energies and Casimir pressures is clarified, and the former are shown to involve surface terms, interpreted as the quantum vacuum energies of the surfaces. The Casimir energy for a δ-function spherical shell in 3 + 1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a δ-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE (δ-function) and TM (derivative of δ-function) Casimir energies. These results clarify recent discussions in the literature.

Lifetimes of a two-dimensional electron gas in the presence of spin-orbit interaction

Using the momentum-balance equation derived from a Boltzmann equation, we evaluate the quantum and transport lifetimes of a two-dimensional electron gas (2DEG) in the presence of spin-orbit interaction (SOI) induced by the Rashba effect. For low temperatures, the electronic transport lifetimes due to impurity scattering differ significantly between different spin branches. The effect is more pronounced for stronger SOI and/or lower electron density. In contrast, the quantum lifetimes depend very weakly on the strength of the SOI and are in line with experimental findings. These results hold for electron-impurity scattering via screened Coulomb or short-range δ-function potentials. The physical reason behind a small difference of the quantum lifetime in different spin branches is discussed.

Transmission through finite 1D barriers containing correlated impurities: an analytic formulation

We address the problem of electronic transmission through a finite sequence of identical potential barriers in one dimension, which has embedded in it “correlated impurities” that is, smaller identical clusters of n-mers, each described by n identical impurity potentials. Analytic formulae are derived for the transmission coefficient, and the investigation of Dunlap et al. [Phys. Rev. Lett. 65 (1990) 88], which were carried out for the corresponding dimer problem in the tight-binding model, are re-examined and generalised. The dependence of the condition for perfect transmission is obtained as a function of the strength of the impurity potential, treated here as a control parameter, in the case of a sequence of δ-function potentials.

Transport in multi-chain models of interacting fermions

We study transport in a class of exactly solvable multi-chain models of interacting fermions using bosonization. We introduce magnetic and non-magnetic impurities at a site, through either weak δ-function potentials or weak links. Using a renormalization group analysis, we show that the strength of the non-magnetic impurity is not affected by the interaction, and hence, conductance through the impurity can be exactly computed. Magnetic impurities, on the other hand, in a single- or two-chain model always cut the wire at the impurity site. However, when the number of chains is greater than two, we find a critical interaction strength beyond which there exists a metal-insulator transition tuned by the strength of the magnetic impurity.

Multifractal scaling of electronic transmission resonances in perfect and imperfect Fibonacci [formula omitted]-function potentials

We present here a detailed multifractal scaling study for the electronic transmission resonances with the system size for an infinitely large one-dimensional perfect and imperfect quasiperiodic system represented by a sequence of δ-function potentials. The electronic transmission resonances in the energy minibands manifest more and more fragmented nature of the transmittance with the change of system size. We claim that when a small perturbation is randomly present at a few number of sites or layers, the nature of electronic states will change and this can be understood by studying the electronic transmittance with the change of system size. We report the different critical states manifested in the size variation of the transmittance corresponding to the resonant energies for both perfect and imperfect cases through multifractal scaling study for few of these resonances.